ESAIM: M2AN 54 (2020) 1917–1949 ESAIM: Mathematical Modelling and Numerical Analysishttps://doi.org/10.1051/m2an/2020025 www.esaim-m2an.org

SOME MULTIPLE FLOW DIRECTION ALGORITHMS FOR OVERLAND FLOWON GENERAL MESHES

Julien Coatleven˚

Abstract. After recalling the most classical multiple flow direction algorithms (MFD), we establishtheir equivalence with a well chosen discretization of Manning–Strickler models for water flow. Fromthis analogy, we derive a new MFD algorithm that remains valid on general, possibly non conformingmeshes. We also derive a convergence theory for MFD algorithms based on the Manning–Stricklermodels. Numerical experiments illustrate the good behavior of the method even on distorted meshes.

Mathematics Subject Classification. 65N08, 65N12, 65N15.

Received September 17, 2019. Accepted April 9, 2020.

1. Introduction

Overland water flow plays a major role in hydrogeology at many time scales, from million of years forstratigraphic studies of interest for the oil and gas industry to several days or weeks for predicting river floodingand landslides. Intermediate time scales are also increasingly studied as they are of crucial interest for climaticforecasts, from glacier withdrawal to desert expansion. Of course, a huge literature exists in the correspondingfields, describing the physical phenomenons involved with an equally huge model diversity (Navier–Stokes,Stokes, shallow water, etc...). Due to the complexity underlying flow models, approximate, phenomenologicallybased models have been developed to increase computational speed. Among those models, a common approachthat has given satisfactory results is the one based on the so-called multiple flow directions algorithms (MFD).

The idea underlying all MFD algorithms is that water routing at large space scales will be mainly governed bythe topographic slope. Using digital elevation models that provide a meshed representation of the topography,those algorithms compute water flow by distributing water from the topographically higher discrete cells tothe topographically lower ones, each distribution formula corresponding to a specific MFD algorithm. In mostcases, the MFD algorithms are developed assuming that the mesh is a uniform cartesian grid, with the samespace step in each direction. Historically, the first MFD algorithms were in fact single flow direction algorithms,where the distribution formula selected only one neighbour, generally the one with the steepest slope. Thedeficiencies of such simple algorithms are the origin of the development of MFD algorithms (see [26]). The mostclassical MFD algorithm [16, 22] uses directly the slope to distribute the flow, while models using powers ofthe slope were developed to concentrate the flow and limit diffusion effects due to the use of coarse meshes

Keywords and phrases. Multiple flow direction algorithm, overland flow, virtual element method, hybrid finite volume, generalmeshes.

IFP Energies nouvelles, 1 et 4 avenue de Bois-Preau, Rueil-Malmaison 92852, France.˚Corresponding author: julien.coatleven@ifpen.fr

Article published by EDP Sciences c EDP Sciences, SMAI 2020

1918 J. COATLEVEN

(see [18, 21, 25]). Some attempts have been made to apply MFD algorithms on more general meshes, with anemphasis on triangular ones (see [24,27]). We refer the reader to [9] for a comparison of some of those methods,and to the references in the aforementioned papers for a more complete overview of the tremendous literatureon MFD algorithms, on which we will not try to be exhaustive.

All those algorithms proceed sequentially from the higher cells to the lower ones, however as was alreadynoticed in [23], using an MFD algorithm is in fact equivalent to solving a linear system using a specific cellordering. The key idea underlying the present paper comes from a deeper study of the linear system associatedwith one of the earliest MFD algorithm. Indeed, we will explain that this linear system is completely equivalent toa two point flux finite volume scheme (see [10]) applied to a stationary Manning–Strickler model for water flow.It was then clear that replacing the TPFA scheme that requires a strong orthogonality hypothesis on meshes toremain valid by more advanced flux approximation schemes would allow us to derive MFD algorithms adaptedto general meshes. Moreover, this equivalence will also allow us to derive a theoretical framework within whichwe will be able to study the convergence properties of MFD algorithms. In our numerical experiments, we haveconsidered two flux reconstructions inspired by two finite volume schemes: the hybrid finite volume (see [11,12])and the virtual volume method (or conservative first order virtual element method, see [5]). Of course, otherchoices could have been made (for instance VAG finite volumes [13,14] or discontinuous Galerkin methods [7]),however those two schemes will be sufficient to illustrate our approach.

The paper will be organized as follows: after describing the data and meshes, we recall the most classical mul-tiple flow direction algorithms, and reformulate them in a more algebraic fashion. Next, using this reformulationwe explain how they are linked with the TPFA scheme for a family of Manning–Strickler models. We elaborateon this basis to overcome the mesh limitations induced by the TPFA scheme, introducing a new family of MFDalgorithms that will remain valid on a huge class of meshes, including those with hanging nodes. We then studythe convergence properties of all methods and conclude by some numerical illustrations.

2. Classical multiple flow direction algorithms

2.1. Mesh and data description

Let Ω be a bounded polyhedral connected domain of R2, whose boundary is denoted BΩ “ ΩzΩ. We recall theusual notations describing a mesh ℳ “ p𝒯 ,ℱq of Ω. 𝒯 is a finite family of connected open disjoint polygonalsubsets of Ω (the cells of the mesh), such that Ω “ Y𝐾P𝒯 𝐾. For any 𝐾 P 𝒯 , we denote by |𝐾| the measure of|𝐾|, by B𝐾 “ 𝐾z𝐾 the boundary of 𝐾, by ℎ𝐾 its diameter and by 𝑥𝐾 its barycenter. ℱ is a finite family ofdisjoint subsets of hyperplanes of R2 included in Ω (the faces of the mesh) such that, for all 𝜎 P ℱ , its measureis denoted |𝜎|, its diameter ℎ𝜎 and its barycenter 𝑥𝜎. For any 𝐾 P 𝒯 , there exists a subset ℱ𝐾 of ℱ such thatB𝐾 “ Y𝜎Pℱ𝐾

𝜎. Then, for any 𝜎 P ℱ , we denote by 𝒯𝜎 “ t𝐾 P 𝒯 | 𝜎 P ℱ𝐾u. Next, for all 𝐾 P 𝒯 and all 𝜎 P ℱ𝐾 ,we denote by 𝑛𝐾,𝜎 the unit normal vector to 𝜎 outward to 𝐾, and 𝑑𝐾,𝜎 “ ||𝑥𝜎´𝑥𝐾 ||. The set of boundary facesis denoted ℱext, while interior faces are denoted ℱint. Finally for any 𝜎 P ℱint, we denote 𝑑𝐾𝐿 “ ||𝑥𝐾 ´ 𝑥𝐿||

where 𝒯𝜎 “ t𝐾, 𝐿u. We assume that there exists a subset ℱext,in of ℱext such that:

BΩin “ď

𝜎Pℱext,in

𝜎 where BΩin “ t𝑥 P BΩ | ∇𝑏 ¨ 𝑛 ą 0u

and we denote of course ℱext,out “ ℱextzℱext,in. As usual, ℎ “ max𝐾P𝒯 ℎ𝐾 will denote the mesh size. In whatfollows, we will assume that our mesh satisfies:

(A1) There exists a real number 𝜌 ą 0 and a matching simplicial submesh 𝒮𝒯 of ℳ such that for any 𝑇 P 𝒮𝒯 ,𝜌ℎ𝑇 ď 𝑟𝑇 where 𝑟𝑇 is the inradius of 𝑇 , and for any 𝑇 P 𝒯 and any 𝑇 P 𝒮𝒯 such that 𝑇 Ă 𝐾, 𝜌ℎ𝐾 ď ℎ𝑇 .

From [3, 7], we know that assumption (A1) implies that for any 𝑘 P N, there exists 𝐶𝑡𝑟,𝑘 ą 0 independent on ℎsuch that for any 𝐾 P 𝒯 , any 𝜎 P ℱ𝐾 and any 𝑝 P P𝑘p𝐾q:

||𝑝||𝐿2p𝜎q ď 𝐶𝑡𝑟,𝑘ℎ´12𝜎 ||𝑝||𝐿2p𝐾q. (2.1)

SOME MULTIPLE FLOW DIRECTION ALGORITHMS 1919

Figure 1. Example of real topography.

There also exists 𝐶𝑡𝑟 ą 0 independent on ℎ such that for any 𝑣 P 𝐻1p𝐾q:

||𝑣||𝐿2p𝜎q ď 𝐶𝑡𝑟

´

ℎ´1𝐾 ||𝑣||2𝐿2p𝐾q ` ℎ𝐾 ||∇𝑣||2𝐿2p𝐾q

¯12

. (2.2)

Finally, assumption (A1) implies that for any integer 𝑘, there exists 𝐶poly,𝑘 ą 0 such that for any 𝐾 P 𝒯 andany 𝑣 P 𝐻𝑠p𝐾q with 𝑠 P t1, . . . , 𝑘 ` 1u, there exists 𝑝 P P𝑘p𝐾q such that

|𝑣 ´ 𝑝|𝐻𝑚p𝐾q ` ℎ12𝐾 |𝑣 ´ 𝑝|𝐻𝑚pB𝐾q ď 𝐶poly,𝑘ℎ𝑠´𝑚

𝐾 |𝑣|𝐻𝑠p𝐾q for 𝑚 P t0, . . . , 𝑠´ 1u . (2.3)

In the estimates that will follow, the constant 𝐶 ą 0 will always denote by convention a quantity independenton the mesh size ℎ, whose value can change from line to line. Also by convention for any 𝜎 P ℱint we willdenote 𝒯𝜎 “ t𝐾, 𝐿u. In the same way, when considering a cell 𝐾 and one of its interior faces 𝜎 P ℱ𝐾 Xℱint, byconvention the cell 𝐿 will denote the other element of 𝒯𝜎.

In order to be able to establish error estimates, we assume that the topography, often called the basement inthe geological community and thus usually denoted 𝑏, satisfies 𝑏 P 𝑊 2,8pΩq and that there exists 𝛼 ą 0 suchthat ´∆𝑏 ě 𝛼 for almost every 𝑥 P Ω. However, from the practical point of view 𝑏 is only known through somepointwise values, very often given as a digitized surface elevation, on which some interpolation and upscaling ordownscaling processes have been applied (see Fig. 1). Thus, its Laplacian cannot be expected to be practicallycomputable, and the above assumption should really be considered as an abstract technical requirement forestablishing convergence estimates. Thus, our data will more realistically consists in some discrete mesh-basedp𝐵𝐾q𝐾P𝒯 representation of 𝑏, where each 𝐵𝐾 can represent several pointwise values of 𝑏, complemented by awater source term 𝑓 P 𝐿8pΩq, possibly also only known through a mesh-based representation.

2.2. The classical multiple flow direction algorithm

In the geological literature, multiple flow direction algorithms are considered as purely algorithmic ways ofdistributing water from one cell to another. Thus, they are generally described in a purely algorithmic fashion.Consequently, in this section to introduce the most classical MFD algorithms we will follow the presentation

1920 J. COATLEVEN

generally found in the geological community, that is to say a purely algorithmic point of view. One of our firsttasks will precisely consists in abstracting ourselves from this algorithmic setting. For any 𝐾 P 𝒯 , let 𝑏𝐾 be avalue for the topography associated with cell 𝐾. To fix notations, consider for the moment that

𝑏𝐾 “1|𝐾|

ż

𝐾

𝑏 @𝐾 P 𝒯

but other approximations can be considered, for instance 𝑏𝐾 “ 𝑏 p𝑥𝐾q for any 𝐾 P 𝒯 if 𝑏 is regular enough.Multiple flow direction algorithms are based on formulae to distribute the water flow from a cell to its neighboursthat are topographically lower. The most classical distribution formula consists simply in distributing the flowproportionally to the ratio 𝑠𝐾𝐿𝑠𝐾 of the discrete slope 𝑠𝐾𝐿 between the high cell 𝐾 and the low cell 𝐿 regardingthe total positive slope 𝑠𝐾 of the high cell 𝐾, where the discrete slope 𝑠𝐾𝐿 is given by

𝑠𝐾𝐿 “|𝜎|

𝑑𝐾𝐿p𝑏𝐾 ´ 𝑏𝐿q

and the total positive slope 𝑠𝐾 of cell 𝐾 is given by

𝑠𝐾 “ÿ

𝜎Pℱ𝐾Xℱint,𝑏𝐾ě𝑏𝐿

|𝜎|

𝑑𝐾𝐿p𝑏𝐾 ´ 𝑏𝐿q .

Notice that in many cases of the literature, as the MFD algorithm is applied on a uniform cartesian mesh withthe same space step in every direction, the face measure |𝜎| is simply omitted (see for instance [16, 22]), withno impact on the ratio 𝑠𝐾𝐿𝑠𝐾 . To give a detailed description of the most classical multiple flow directionalgorithms, we need to introduce some notations. Let 𝑏0 “ max t𝑏𝐾 | 𝐾 P 𝒯 u, 𝒯0 “ t𝐾 P 𝒯 | 𝑏𝐾 “ 𝑏0u and𝒯´1 “ H. The set 𝒯0 thus denotes the set of cells with maximum topographic height. Then, for any 𝑛 P N wedefine the set 𝒯𝑛 of elements of 𝒯 by setting:

𝒯𝑛 “ď

0ď𝑖ď𝑛

𝒯𝑖 where 𝒯𝑖 “ t𝐾 P 𝒯 | 𝑏𝐾 “ 𝑏𝑖u and 𝑏𝑖 “ max!

𝑏𝐾 | 𝐾 P 𝒯 z𝒯𝑖´1

)

.

By construction, as 𝒯 is a finite set there exists 𝑁𝑏 ą 0 such that 𝒯𝑁𝑏´1 ‰ H and 𝒯𝑁𝑏´1 “ 𝒯 . Thus, for any𝑛 ě 𝑁𝑏, we have 𝒯𝑛 “ H and 𝒯𝑛 “ 𝒯𝑁𝑏´1.

To model water sources, we define a family p𝑓𝐾q𝐾P𝒯 by setting:

𝑓𝐾 “1|𝐾|

ż

𝐾

𝑓 @𝐾 P 𝒯 .

The classical MFD algorithm (reformulated from [16,22]) then reads as follows (see Fig. 2 for a visual illustration,where the width of the arrows is roughly following the amount of distributed water):

(i) For any 𝐾 P 𝒯 , the water influx r𝑞𝐾 is initialized at 0.

(ii) For any 𝐾 P 𝒯0, the water influx r𝑞𝐾 is given by r𝑞𝐾 “ |𝐾|𝑓𝐾 .

(iii) Loop for 𝑛 “ 1 . . . 𝑁𝑏 ´ 1.

(iii-1) For any 𝐿 P 𝒯𝑛´1, we distribute the entire water influx r𝑞𝐿 to the neighbours that belong to 𝒯𝑛 propor-tionally to the slope between the cell 𝐿 and its neighbour, i.e.:

r𝑞𝐾 ÐÝ r𝑞𝐾 `ÿ

𝜎Pℱ𝐾Xℱint,𝑏𝐾ă𝑏𝐿,𝐿P𝒯𝑛´1

|𝜎|r𝑞𝐿

𝑑𝐾𝐿𝑠𝐿p𝑏𝐿 ´ 𝑏𝐾q for all 𝐾 P 𝒯𝑛

SOME MULTIPLE FLOW DIRECTION ALGORITHMS 1921

Figure 2. Basic principle of MFD algorithms: water is distributed to lower neighbouring cellsproportionally to the slope.

where 𝑠𝐿 is the total positive slope in cell 𝐿, and ÐÝ denotes the action of updating r𝑞𝐾 .

(iii-2) For any 𝐾 P 𝒯𝑛, the water influx is complemented by the local sources by setting

r𝑞𝐾 ÐÝ r𝑞𝐾 ` |𝐾|𝑓𝐾 .

(iv) End loop for 𝑛 “ 1 . . . 𝑁𝑏 ´ 1.The MFD algorithm admits a reformulation as a linear system that will play a key role in the remaining of

the paper. To our knowledge, only [23] mentioned this reformulation, although without exhibiting an explicitformula. This link seemed to have been most of the time simply overlooked by the geological community:

Theorem 2.1. The MFD algorithm is equivalent to solving the following linear system for the unknownpr𝑞𝐾q𝐾P𝒯

r𝑞𝐾 ´ÿ

𝜎Pℱ𝐾Xℱint,𝑏𝐾ă𝑏𝐿

|𝜎|r𝑞𝐿

𝑑𝐾𝐿𝑠𝐿p𝑏𝐿 ´ 𝑏𝐾q “ |𝐾|𝑓𝐾 @𝐾 P 𝒯 (2.4)

where

𝑠𝐾 “ÿ

𝜎Pℱ𝐾Xℱint,𝑏𝐾ě𝑏𝐿

|𝜎|

𝑑𝐾𝐿p𝑏𝐾 ´ 𝑏𝐿q

using an ordering for the cells of 𝒯 based on decreasing topography 𝑏𝐾 and a lower triangular solver.

Proof. First, notice that as cells 𝐾 P 𝒯𝑛 are updated only at step 𝑛, their expression is complete past this stepand given by:

r𝑞𝐾 “ÿ

𝜎Pℱ𝐾Xℱint,𝑏𝐾ă𝑏𝐿,𝐿P𝒯𝑛´1

|𝜎|r𝑞𝐿

𝑑𝐾𝐿𝑠𝐿p𝑏𝐿 ´ 𝑏𝐾q ` |𝐾|𝑓𝐾 for all 𝐾 P 𝒯 z𝒯𝑛.

1922 J. COATLEVEN

However, by construction of the sets 𝒯𝑖, any 𝐿 P 𝒯 such that 𝑏𝐿 ą 𝑏𝐾 for 𝐾 P 𝒯𝑛, will satisfy 𝐿 P 𝒯𝑛´1. Thus,the above expression can be simplified in:

r𝑞𝐾 “ÿ

𝜎Pℱ𝐾Xℱint,𝑏𝐾ă𝑏𝐿

|𝜎|r𝑞𝐿

𝑑𝐾𝐿𝑠𝐿p𝑏𝐿 ´ 𝑏𝐾q ` |𝐾|𝑓𝐾 for all 𝐾 P 𝒯 z𝒯𝑛

where the set 𝒯𝑛´1 no longer appears. As for any 𝐾 P 𝒯 , there exists 0 ď 𝑛 ď 𝑁𝑏 ´ 1 such that 𝐾 P 𝒯𝑛,and noticing that 𝑠𝐿 ą 0 as soon as there exists 𝜎 P ℱ𝐿 X ℱint, 𝒯𝜎 “ t𝐾, 𝐿u such that 𝑏𝐿 ą 𝑏𝐾 , (2.4) followsby induction. Finally, starting from (2.4), if one chooses an ordering for the cells of 𝒯 based on decreasingtopography 𝑏𝐾 , it is clear that the above system becomes a lower triangular one for r𝑞𝐾 . It should be obvi-ous at this point that the associated lower triangular solver then coincides exactly with the classical MFDalgorithm.

A great advantage of considering the above system rather than its algorithmic counterpart is that it makesclear that triangular solvers are not the only possible linear solvers, which was indeed the main point of [23].A wide range of linear solvers can be used to solve (2.4), and most importantly parallel solvers than canconsiderably speed up the solving process on meshes with a huge cell number.

Remark 2.2. Many variants of this classical MFD algorithm exist, which at least to the authors knowledgemainly consist in modifying the way the influx is distributed from an upper cell to its lower neighbours: powersof the slope instead of the slope, probabilistic repartitions, repartitions using a specified number of neighboursfor a given mesh structure, etc... With the exception of distribution formulae that use powers of the slope insteadof the slope itself, proceeding as we have done for the most classical MFD algorithm it is not difficult to showthat all those variants can be rewritten under the form:

r𝑞𝐾 ´ÿ

𝜎Pℱ𝐾Xℱint,𝑏𝐾ă𝑏𝐿

𝜆𝜎|𝜎|r𝑞𝐿

𝑑𝐾𝐿𝑠𝐿p𝑏𝐿 ´ 𝑏𝐾q “ |𝐾|𝑓𝐾 @𝐾 P 𝒯

with this time

𝑠𝐾 “ÿ

𝜎Pℱ𝐾Xℱint,𝑏𝐾ě𝑏𝐿

𝜆𝜎|𝜎|

𝑑𝐾𝐿p𝑏𝐾 ´ 𝑏𝐿q .

The coefficient 𝜆𝜎 associated with each face will take different values depending on the way one want to modifythe influx repartition. Notice that we have chosen to not consider distribution formulae based on powers of theslope as they only had some technical difficulties with no fundamental differences in the analysis.

3. On the link with Manning–Strickler models

Consider the following classical transient Manning–Strickler model:ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

B𝑢

B𝑡´ div p𝜆𝑢𝑚∇𝑏q “ 𝑓 in Ω

´𝑢𝑚∇𝑏 ¨ 𝑛 “ 0 on BΩin

where 𝑢 is the water height, 𝑚 a parameter and 𝑛 denotes the outward normal to Ω. The coefficient 𝜆 P 𝐿8pΩqis the inverse of the Manning–Strickler coefficient expressing soil roughness, and is assumed to satisfy 0 ă 𝜆´ ď𝜆 ď 𝜆` ă `8 almost everywhere in Ω. Formally, the steady state associated with the above system is thefollowing stationary Manning–Strickler model for overland flow:

ˇ

ˇ

ˇ

ˇ

ˇ

´div p𝜆𝑢𝑚∇𝑏q “ 𝑓 in Ω

´𝜆𝑢𝑚∇𝑏 ¨ 𝑛 “ 0 on BΩin.(3.1)

SOME MULTIPLE FLOW DIRECTION ALGORITHMS 1923

Denoting 𝑢𝜆,𝑚 “ 𝜆𝑢𝑚 , 𝛽 “ ´∇𝑏 P 𝑊 1,8pΩq and 𝜇 “ ´∆𝑏 ě 𝛼 ą 0 P 𝐿8pΩq, remark that (3.1) can berewritten:

ˇ

ˇ

ˇ

ˇ

ˇ

𝛽 ¨∇𝑢𝜆,𝑚 ` 𝜇𝑢𝜆,𝑚 “ 𝑓 in Ω

𝑢𝜆,𝑚 “ 0 on BΩin.(3.2)

Stationary transport problems of the form (3.2) have of course received a lot of attention in the existingliterature, in particular as they correspond to a popular model for neutron transport. Their well-posedness hasbeen considered for instance in [1, 2], or more recently in [8, 15, 17]. Thus, from the results of [8, 15] and theregularity hypotheses on 𝑏, 𝑓 , and 𝜆, we know that there exists a unique 𝑢 P 𝐿8pΩq solution of (3.2). As thewater height 𝑢 only appears through its 𝑚-th power 𝑢𝑚, in the remaining of the paper we will with a slightabuse of notations directly use 𝑢 instead of 𝑢𝑚.

We are now going to explain that the classical MFD algorithm coincides with a well chosen discretizationof the stationary Manning–Strickler model. Assume that the mesh is orthogonal, i.e. there exists a family ofcentroids p𝑥𝐾q𝐾P𝒯 such that:

𝑥𝐾 P 𝐾 @𝐾 P 𝒯 and𝑥𝐿 ´ 𝑥𝐾

||𝑥𝐿 ´ 𝑥𝐾 ||“ 𝑛𝐾,𝜎 for 𝜎 P ℱint, 𝜎 “ t𝐾, 𝐿u

and let us denote 𝑑𝐾,𝜎 the distance of 𝑥𝐾 to the hyperplane containing 𝜎 for any 𝜎 P ℱ𝐾 and any 𝐾 P 𝒯 .Then, one can use a two-point finite volume scheme to discretize the steady-state Manning model. Assume that𝑏𝐾 “ 𝑏 p𝑥𝐾q or at least a second order approximation of it. Denoting 𝑢𝐾 for any 𝐾 P 𝒯 the discrete waterheight unknown, if one further assumes that 𝑏𝜎 “ 𝑏𝐾 for any 𝜎 P ℱext and 𝐾 P 𝒯𝜎 which is generally what isdone in practical applications of the MFD algorithm, for any 𝐾 P 𝒯 we get:

ÿ

𝜎Pℱ𝐾Xℱint

𝜏𝐾𝐿r𝑢up𝜎 p𝑏𝐾 ´ 𝑏𝐿q “ |𝐾|𝑓𝐾

where r𝑢up𝜎 “ 𝑢𝐾 if 𝑏𝐾 ě 𝑏𝐿 and r𝑢up

𝜎 “ 𝑢𝐿 if 𝑏𝐾 ă 𝑏𝐿 and the transmissivity 𝜏𝐾𝐿 is given for instance by theharmonic mean:

𝜏𝐾𝐿 “|𝜎|𝜆𝐾𝜆𝐿

𝜆𝐾𝑑𝐿,𝜎 ` 𝜆𝐿𝑑𝐾,𝜎

where 𝜆𝐾 “1|𝐾|

ż

𝐾

𝜆.

Immediately we deduce the following equivalence result, which despite its simplicity is the cornerstone of thepresent paper:

Theorem 3.1. The TPFA scheme for Manning Strickler’s model is equivalent to the MFD algorithm.

Proof. Gathering the faces by upwinding kind, we get:ÿ

𝜎Pℱ𝐾Xℱint,𝑏𝐾ě𝑏𝐿

𝜏𝐾𝐿𝑢𝐾 p𝑏𝐾 ´ 𝑏𝐿q ´ÿ

𝜎Pℱ𝐾Xℱint,𝑏𝐾ă𝑏𝐿

𝜏𝐾𝐿𝑢𝐿 p𝑏𝐿 ´ 𝑏𝐾q “ |𝐾|𝑓𝐾 . (3.3)

Setting𝑠𝐾 “

ÿ

𝜎Pℱ𝐾Xℱint,𝑏𝐾ě𝑏𝐿

𝜏𝐾𝐿 p𝑏𝐾 ´ 𝑏𝐿q

and noticing that 𝑠𝐿 ą 0 as soon as there exists 𝜎 P ℱ𝐿 X ℱint such that 𝑏𝐿 ą 𝑏𝐾 , we see that equation (3.3)can be rewritten:

𝑠𝐾𝑢𝐾 ´ÿ

𝜎Pℱ𝐾Xℱint,𝑏𝐾ă𝑏𝐿

𝜏𝐾𝐿𝑢𝐿 p𝑏𝐿 ´ 𝑏𝐾q “ |𝐾|𝑓𝐾 .

Defining the water influx by r𝑞𝐾 “ 𝑠𝐾𝑢𝐾 , we thus obtain:

r𝑞𝐾 ´ÿ

𝜎Pℱ𝐾Xℱint,𝑏𝐾ă𝑏𝐿

𝜏𝐾𝐿r𝑞𝐿

𝑠𝐿p𝑏𝐿 ´ 𝑏𝐾q “ |𝐾|𝑓𝐾 . (3.4)

If we take 𝜆 “ 1, immediately we see that (3.4) is exactly the MFD equation (2.4).

1924 J. COATLEVEN

Surprisingly, this result seems to be absent of the MFD literature. The main reason is probably that formu-lations of MFD algorithms as linear systems are equally difficult to find. From this equivalence between theclassical MFD and the two-point flux approximation (TPFA) of the classical Manning–Strickler model, someuseful observations can be made. Probably the most surprising one is that the MFD unknown pr𝑞𝐾q𝐾P𝒯 can beused instead of 𝑢𝐾 to solve the two-point approximation of the Manning–Strickler model. Moreover, existenceand uniqueness of solutions of the MFD problem (3.4) are now an immediate consequence of its equivalencewith the MFD algorithm without requiring any hypothesis on the topography. Next, let us mention that whenmodeling overland flow, the quantity of interest is not the water height but the water discharge, i.e. the norm||𝜆ℎ𝑚∇𝑏|| of the water flux vector. This is the reason why no water height explicitly appears in MFD algorithms.However, a crucial consequence of our equivalence result is that the usual unknown r𝑞𝐾 of the MFD algorithm,while an excellent choice from the algebraic perspective, is probably not the good quantity to represent the waterdischarge. Indeed, using r𝑞𝐾 “ 𝑠𝐾𝑢𝐾 and the consistency of the two-point formula we see that it approximates:

r𝑞𝐾 «ÿ

𝜎Pℱ𝐾

ż

𝜎

𝑢 p´𝜆∇𝑏 ¨ 𝑛𝐾,𝜎q`

where 𝑣` “ maxp0, 𝑣q. We cannot expect such a r𝑞𝐾 to be an approximation of the norm of 𝜆𝑢∇𝑏, as itapproximates the accumulated influx in a cell which is a mesh dependent quantity. Thus, no kind of convergencelet alone approximation properties can be expected for such a quantity in general. To effectively compute adiscrete water discharge 𝑞𝐾 for each cell 𝐾 P 𝒯 , we reconstruct cellwise the water flux vector by setting:

𝑄𝐾 “ÿ

𝜎Pℱ𝐾Xℱint,𝑏𝐾ą𝑏𝐿

𝜏𝐾𝐿r𝑞𝐾

|𝐾|𝑠𝐾p𝑏𝐾 ´ 𝑏𝐿q p𝑥𝜎 ´ 𝑥𝐾q ´

ÿ

𝜎Pℱ𝐾Xℱint,𝑏𝐾ă𝑏𝐿

𝜏𝐾𝐿r𝑞𝐿

|𝐾|𝑠𝐿p𝑏𝐿 ´ 𝑏𝐾q p𝑥𝜎 ´ 𝑥𝐾q .

The consistent water discharge is immediately given by 𝑞𝐾 “ ||𝑄𝐾 ||. We will illustrate in the numerical sectionthe convergence deficiency of r𝑞𝐾 , and how 𝑞𝐾 has a much better behavior. Obviously, for any 𝐾 P 𝒯 such that𝑠𝐾 ‰ 0, we can compute an equivalent positive water height by setting 𝑢𝐾 “ r𝑞𝐾𝑠𝐾 . Cells where 𝑠𝐾 “ 0 arecells where all discrete fluxes are ingoing fluxes, thus it is clear that one should either set 𝑢𝐾 “ `8 or 𝑢𝐾 “ 0depending if water effectively reaches the cell or not. From the definition of 𝑄𝐾 , we see that the value of 𝑢𝐾

for such cells will have no influence on the water discharge and its asymptotic convergence. It is thus clear thatone can always define 𝑢𝐾 by setting:

𝑢𝐾 “

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

r𝑞𝐾

𝑠𝐾if 𝑠𝐾 ą 0

0 otherwise.

Remark 3.2. Notice that using the harmonic mean is not mandatory if one can use an approximation of 𝜆directly on the faces, leading to the transmissivity 𝜏𝐾𝐿 “ |𝜎|𝜆𝜎𝑑𝐾𝐿 in which case (3.4) will correspond to oneof the many variants of the classical MFD.

4. Some conservative and consistent flux reconstructions on general meshes

From the equivalence between the MFD algorithms and the TPFA scheme for the stationary Manning–Strickler model, an obvious way of generalizing MFD algorithms to general meshes consists in replacing theTPFA flux reconstruction formula by more advanced flux reconstruction techniques that are still valid ongeneral meshes. We follow the idea of [4]: we consider a gradient reconstruction in each cell that will consist ina consistent part plus a stabilization part. However, contrary to [4] where the stabilization was used to obtaincoercivity, here we will use this stabilization to enforce conservativity.

SOME MULTIPLE FLOW DIRECTION ALGORITHMS 1925

4.1. A conservative flux reconstruction formula

For any cell 𝐾 P 𝒯 , let us denote 𝐵𝐾 “ 𝒟𝐾p𝑏q the local values associated with the topography 𝑏, and𝑋𝐾 “ R𝒩𝐾 the set of local values, with 𝒩𝐾 the number of local values. The operator 𝒟𝐾 : 𝐻1p𝐾q ÞÝÑ 𝑋𝐾

will of course depend on the considered reconstruction formula, however to simplify notations we consider thatthe value 𝑏𝐾 at 𝑥𝐾 always belongs to the set of local values. Those local values represent all the knowledge wehave in practice of the field 𝑏, and once again its Laplacian cannot be expected to be computable. For typicalapplications such as stratigraphic modelling it consists in cell values complemented by vertex or face values, thusconditioning the schemes we can effectively use. We assume that we are given a gradient reconstruction operator∇𝐾 : 𝑋𝐾 ÞÝÑ R2, and we define a reconstruction formula in each cell through the operator Π𝐾 : 𝑋𝐾 ÞÝÑ P1p𝐾q,where P1p𝐾q is the set of first order polynomials on 𝐾 and:

Π𝐾 p𝐵𝐾q p𝑥q “ 𝑏𝐾 `∇𝐾 p𝐵𝐾q ¨ p𝑥´ 𝑥𝐾q

and thus ∇Π𝐾 p𝐵𝐾q “ ∇𝐾 p𝐵𝐾q. Such gradient reconstructions are the usual basis of modern finite volumemethods (see [4, 5, 12]), and many formulae can be found in the literature. Using those reconstructions, it istempting to define the flux operator 𝐹𝐾,𝜎 p𝐵𝐾q by setting:

𝐹𝐾,𝜎 p𝐵𝐾q “ ´|𝜎|𝜆𝐾∇𝐾 p𝐵𝐾q ¨ 𝑛𝐾,𝜎.

As consistency of the flux will of course rely on the consistency of the gradient reconstruction operators, theabove definition will indeed lead to consistent fluxes, however to construct our generalized MFD algorithmsconservativity will play a crucial role. This means that we will require the conservativity of the flux:

ÿ

𝐾P𝒯𝜎

𝐹𝐾,𝜎 p𝐵𝐾q “ 0

which cannot be satisfied in general (except by the two-point fluxes) with such a simple formula. This is thereason why, following the ideas of [4], we introduce a stabilized gradient operator ∇𝐾,𝜎 : 𝑋𝐾 ˆ R ÞÝÑ R bysetting:

∇𝐾,𝜎 p𝐵𝐾 , 𝜌𝜎q “ ∇𝐾 p𝐵𝐾q `𝜂

𝑑𝐾,𝜎pp𝜌𝜎 ´ 𝑏𝐾q ´∇𝐾 p𝐵𝐾q ¨ p𝑥𝜎 ´ 𝑥𝐾qq𝑛𝐾,𝜎

where 𝜂 ą 0 is a stabilization parameter. Then, for any 𝐾 P 𝒯 and any 𝜎 P ℱ𝐾 , we define an intermediate fluxoperator 𝐹𝐾,𝜎 : 𝑋𝐾 ˆ R ÞÝÑ R by setting:

𝐹𝐾,𝜎 p𝐵𝐾 , 𝜌𝜎q “ ´|𝜎|𝜆𝐾∇𝐾,𝜎 p𝐵𝐾 , 𝜌𝜎q ¨ 𝑛𝐾,𝜎.

Obviously, for any 𝜎 P ℱint, we would like to define 𝜎 as the solution ofÿ

𝐾P𝒯𝜎

𝐹𝐾,𝜎

´

𝐵𝐾 , 𝜎

¯

“ 0. (4.1)

Fortunately, as 𝜆 is positive almost everywhere we immediately deduce that such a 𝜎 is always defined and isgiven by:

𝜎 “1

˜

ÿ

𝑀P𝒯𝜎

𝜂𝜆𝑀

𝑑𝑀,𝜎

¸

˜

ÿ

𝐾P𝒯𝜎

𝜂𝜆𝐾

𝑑𝐾,𝜎𝑏𝐾 `

ÿ

𝐾P𝒯𝜎

∇𝐾 p𝐵𝐾q ¨

ˆ

𝜂𝜆𝐾

𝑑𝐾,𝜎p𝑥𝜎 ´ 𝑥𝐾q ´ 𝜆𝐾𝑛𝐾,𝜎

˙

¸

. (4.2)

Thus, for any 𝜎 P ℱint, it is legitimate to define the conservative flux operator 𝐹𝐾,𝜎 :Ś

𝐿P𝒯𝜎𝑋𝐿 ÞÝÑ R by

setting:𝐹𝐾,𝜎 p𝐵𝜎q “ 𝐹𝐾,𝜎

´

𝐵𝐾 , 𝜎

¯

(4.3)

1926 J. COATLEVEN

where 𝜎 is the unique solution of (4.1), while for any 𝜎 P ℱext, we simply set:

𝐹𝐾,𝜎 p𝐵𝜎q “ ´|𝜎|𝜆𝐾∇𝐾 p𝐵𝐾q ¨ 𝑛𝐾,𝜎. (4.4)

Remark 4.1. It is tempting to recover conservativity using the following simpler formula for 𝐹𝐾,𝜎 p𝐵𝜎q:

𝐹𝐾,𝜎 p𝐵𝜎q “12p𝐹𝐾,𝜎 p𝐵𝐾q ´ 𝐹𝐿,𝜎 p𝐵𝐿qq .

The main drawback of this alternative formula can be understood in cases where 𝜆 is discontinuous, where thiswould obviously lead to a very coarse approximation of the flux along the lines of discontinuity of 𝜆. Providedthose discontinuities are resolved by the mesh, our formula for 𝐹𝐾,𝜎 p𝐵𝜎q will instead behave as the usualharmonic mean. It is thus a more robust and versatile choice.

4.2. Consistency properties of the flux

Denoting 𝐵p𝑥, 𝜉q the ball centered at 𝑥 of radius 𝜉 and 𝐵Ωp𝑥, 𝜉q “ 𝐵p𝑥, 𝜉qXΩ, we recall the usual definitionof strong consistency for gradient reconstruction operators:

Definition 4.2 (Strong consistency). The gradient reconstruction operator ∇𝐾 is strongly consistent if andonly if there exists 𝐶 ą 0 independent on ℎ such that for any 𝜙 P 𝐶2

´

𝐵Ω p𝑥𝐾 , 𝜉q¯

, every 0 ă 𝜉 ď 2ℎ𝐾 andevery 𝑥 P 𝐵Ω p𝑥𝐾 , 𝜉q:

|∇𝜙p𝑥q ´∇𝐾 p𝒟𝐾p𝜙qq| ď 𝐶𝜉||𝜙||𝑊 2,8p𝐵Ωp𝑥𝐾 ,𝜉qXΩq.

As an immediate consequence of the above definition, we have, using Taylor’s expansion and the density of𝐶8

´

𝐵Ω p𝑥𝐾 , 𝜉q¯

in 𝑊 2,8 p𝐵Ωp𝑥, 𝜉qq that for any 𝜙 P 𝑊 2,8pΩq there exists 𝐶 ą 0 depending on 𝜙 but not onℎ such that for every ℎ𝐾 ď 𝜉 ď 2ℎ𝐾 , any 𝐾 P 𝒯 and almost every 𝑥 P 𝐵Ω p𝑥𝐾 , 𝜉q:

|∇𝜙p𝑥q ´∇𝐾 p𝒟𝐾p𝜙qq| ď 𝐶𝜉||𝜙||𝑊 2,8p𝐵Ωp𝑥𝐾 ,𝜉qq

and|𝜙p𝑥q ´Π𝐾 p𝒟𝐾p𝜙qq p𝑥q| ď 𝐶𝜉2||𝜙||𝑊 2,8p𝐵Ωp𝑥𝐾 ,𝜉qq.

Another immediate consequence is the consistency of our discrete fluxes:

Proposition 4.3. Assume that assumption (A1) is satisfied and that the gradient reconstruction operator isstrongly consistent. Then, there exists 𝐶 ą 0 independent on ℎ such that for any 𝐾 P 𝒯 , any 𝜎 P ℱ𝐾 , any𝑏 P 𝑊 2,8pΩq and any 𝜆 P 𝑊 1,8pΩq, we have:

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

1|𝜎|

𝐹𝐾,𝜎 p𝐵𝜎q ` 𝜆∇𝑏 ¨ 𝑛𝐾,𝜎

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

𝐿8p𝐵Ωp𝑥𝐾 ,2ℎ𝐾qq

ď 𝐶ℎ𝐾 ||𝜆||𝑊 1,8pΩq||𝑏||𝑊 2,8pΩq. (4.5)

Proof. By density of 𝐶8`

Ω˘

in 𝑊 1,8pΩq and 𝑊 2,8pΩq, it is clear that it suffices to establish the result for𝜆 P 𝐶8

`

Ω˘

and 𝑏 P 𝐶8`

Ω˘

. Assuming such regularity, we start by establishing that there exists 𝐶 ą 0independent on ℎ such that for any 𝜎 P ℱint:

|𝜎 ´ 𝑏 p𝑥𝜎q | ď 𝐶ℎ2𝐾 ||𝑏||𝑊 2,8pΩq and |𝜎 ´Π𝐾 p𝐵𝐾q p𝑥𝜎q | ď 𝐶ℎ2

𝐾 ||𝑏||𝑊 2,8pΩq @𝐾 P 𝒯𝜎. (4.6)

As by construction, we have Π𝐾 p𝐵𝐾q p𝑥𝜎q “ 𝑏𝐾 `∇𝐾 p𝐵𝐾q ¨ p𝑥𝜎 ´ 𝑥𝐾q, slightly rewriting (4.2) we get:

𝜎 ´ 𝑏 p𝑥𝜎q “1

˜

ÿ

𝑀P𝒯𝜎

𝜂𝜆𝑀

𝑑𝑀,𝜎

¸

˜

ÿ

𝐿P𝒯𝜎

𝜂𝜆𝐿

𝑑𝐿,𝜎pΠ𝐿 p𝐵𝐿q p𝑥𝜎q ´ 𝑏 p𝑥𝜎qq ´

ÿ

𝐿P𝒯𝜎

𝜆𝐿∇𝐿 p𝐵𝐿q ¨ 𝑛𝐿,𝜎

¸

.

SOME MULTIPLE FLOW DIRECTION ALGORITHMS 1927

Immediately, using the strong consistency of the gradient operator we get that for all 𝐾 P 𝒯𝜎:

|Π𝐾 p𝐵𝐾q p𝑥𝜎q ´ 𝑏 p𝑥𝜎q | ď 𝐶ℎ2𝐾 ||𝑏||𝑊 2,8p𝐵Ωp𝑥𝐾 ,ℎ𝐾qq

and thus for all 𝐾 P 𝒯𝜎:

1˜

ÿ

𝑀P𝒯𝜎

𝜂𝜆𝑀

𝑑𝑀,𝜎

¸

ÿ

𝐿P𝒯𝜎

𝜂𝜆𝐿

𝑑𝐿,𝜎|Π𝐿 p𝐵𝐿q p𝑥𝜎q ´ 𝑏 p𝑥𝜎q | ď 𝐶

ÿ

𝐿P𝒯𝜎

𝜂𝜆𝐿

𝑑𝐿,𝜎ℎ2

𝐿

˜

ÿ

𝑀P𝒯𝜎

𝜂𝜆𝑀

𝑑𝑀,𝜎

¸ ||𝑏||𝑊 2,8p𝐵Ωp𝑥𝐾 ,ℎ𝐾qq ď 𝐶ℎ2𝐾 ||𝑏||𝑊 2,8pΩq

as under assumption (A1), we know (see [3,7]) that there exists 𝐶 ą 0 such that for any p𝐾, 𝑀q P 𝒯 2 such thatℱ𝐾 Xℱ𝑀 ‰ H, we have ℎ𝑀ℎ𝐾 ď 𝐶. Moreover, as the gradient operator is strongly consistent, we know that:

|𝜆𝐾∇𝐾 p𝐵𝐾q ´ 𝜆 p𝑥𝜎q∇𝑏 p𝑥𝜎q | ď 𝐶ℎ`

||𝜆||𝑊 1,8pΩq||𝑏||𝑊 1,8pΩq ` ||𝜆||𝐿8pΩq||𝑏||𝑊 2,8pΩq

˘

.

Using:ÿ

𝐾P𝒯𝜎

𝜆 p𝑥𝜎q∇𝑏 p𝑥𝜎q ¨ 𝑛𝐾,𝜎 “ 0

we get:ˇ

ˇ

ˇ

ˇ

ˇ

ÿ

𝐾P𝒯𝜎

𝜆𝐾∇𝐾 p𝐵𝐾q ¨ 𝑛𝐾,𝜎

ˇ

ˇ

ˇ

ˇ

ˇ

ď 𝐶ℎ`

||𝜆||𝑊 1,8pΩq||𝑏||𝑊 1,8pΩq ` ||𝜆||𝐿8pΩq||𝑏||𝑊 2,8pΩq

˘

.

Meanwhile, we clearly get using again assumption (A1):

1˜

ÿ

𝑀P𝒯𝜎

𝜂𝜆𝑀

𝑑𝑀,𝜎

¸ ď1

𝜂𝜆´

1˜

ÿ

𝑀P𝒯𝜎

1𝑑𝑀,𝜎

¸ ď1

𝜂𝜆´card p𝒯𝜎qmax𝑀P𝒯𝜎

𝑑𝑀,𝜎 ď1

𝜂𝜆´max𝑀P𝒯𝜎

ℎ𝑀 ď𝐶ℎ𝐾

𝜂𝜆´

which finally establishes that:|𝜎 ´ 𝑏 p𝑥𝜎q | ď 𝐶ℎ2

𝐾 ||𝑏||𝑊 2,8pΩq

and (4.6) immediately follows using the triangular inequality and the strong consistency of the gradient operator.Then, for any 𝑥 P 𝐵Ω p𝑥𝐾 , 2ℎ𝐾q and any 𝜎 P ℱint:

1|𝜎|

𝐹𝐾,𝜎 p𝐵𝜎q ` 𝜆p𝑥q∇𝑏p𝑥q ¨ 𝑛𝐾,𝜎 “ 𝜆𝐾p∇𝑏p𝑥q ´∇𝐾 p𝐵𝐾qq ¨ 𝑛𝐾,𝜎 ` p𝜆p𝑥q ´ 𝜆𝐾q∇𝑏p𝑥q ¨ 𝑛𝐾,𝜎

´𝜂𝜆𝐾

𝑑𝐾,𝜎

´

𝜎 ´Π𝐾 p𝐵𝐾q p𝑥𝜎q

¯

.

Immediately, as the gradient reconstruction operator is strongly consistent we get:

|𝜆𝐾 p∇𝑏p𝑥q ´∇𝐾 p𝐵𝐾qq ¨ 𝑛𝐾,𝜎| ď 𝐶𝜆`ℎ𝐾 ||𝑏||𝑊 2,8pΩq

while Taylor’s expansion immediately gives:

|p𝜆p𝑥q ´ 𝜆𝐾q∇𝑏p𝑥q ¨ 𝑛𝐾,𝜎| ď 𝐶ℎ𝐾 ||𝜆||𝑊 1,8pΩq||𝑏||𝑊 1,8 .

Under assumption (A1), we have ℎ𝐾𝑑𝐾,𝜎 ď ℎ𝐾𝜌𝐾 ď 1𝜌 which combined with (4.6) gives the desired resultfor 𝜎 P ℱint. For 𝜎 P ℱext, we directly get:

1|𝜎|

𝐹𝐾,𝜎 p𝐵𝜎q ` 𝜆p𝑥q∇𝑏p𝑥q ¨ 𝑛𝐾,𝜎 “ 𝜆𝐾 p∇𝑏p𝑥q ´∇𝐾 p𝐵𝐾qq ¨ 𝑛𝐾,𝜎 ` p𝜆p𝑥q ´ 𝜆𝐾q∇𝑏p𝑥q ¨ 𝑛𝐾,𝜎

and thus applying the same estimates than in the case of interior faces concludes the proof of (4.5).

1928 J. COATLEVEN

5. A multiple flow direction algorithm on general meshes

5.1. The generalized MFD algorithm

Given an approximation p𝐵𝐾q𝐾P𝒯 of the topography and p𝑓𝐾q𝐾P𝒯 of the source term as data, and usingboth upwinding for the water height and our discrete fluxes, the most natural discrete formulation of theManning–Strickler model consists in finding

´

p𝑢𝐾q𝐾P𝒯 , p𝑢𝜎q𝜎Pℱ𝒟ext,in

¯

such that:

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ÿ

𝜎Pℱ𝐾

𝑢up𝜎 𝐹𝐾,𝜎 p𝐵𝜎q “ |𝐾|𝑓𝐾 @𝐾 P 𝒯

𝑢𝜎 “ 0 @𝜎 P ℱ𝒟ext,in

(5.1)

where the upwinding formula 𝑣up𝜎 for any set of cell values p𝑣𝐾q𝐾P𝒯 is defined by:

𝑣up𝜎 “

ˇ

ˇ

ˇ

ˇ

ˇ

𝑣𝐾 if 𝐹𝐾,𝜎 p𝐵𝜎q ě 0

𝑣𝐿 if 𝐹𝐾,𝜎 p𝐵𝜎q ă 0@𝜎 P ℱint, 𝒯𝜎 “ t𝐾, 𝐿u

and

𝑣up𝜎 “

ˇ

ˇ

ˇ

ˇ

ˇ

𝑣𝐾 if 𝐹𝐾,𝜎 p𝐵𝜎q ě 0

0 if 𝐹𝐾,𝜎 p𝐵𝜎q ă 0@𝜎 P ℱext, 𝒯𝜎 “ t𝐾u

and where we have defined the influx boundary associated with the discrete fluxes by setting

ℱ𝒟ext,in “ t𝜎 P ℱext | 𝐹𝐾,𝜎 p𝐵𝜎q ă 0, 𝒯𝜎 “ t𝐾u u .

By construction of the fluxes, for any 𝜎 P ℱint we have that 𝐹𝐿,𝜎 p𝐵𝜎q “ ´𝐹𝐾,𝜎 p𝐵𝜎q. Mimicking what we havedone for the TPFA scheme, we gather faces by upwinding kind and use the fact that 𝑢up

𝜎 “ 0 for all 𝜎 P ℱ𝒟ext,in,leading to:

¨

˝

ÿ

𝜎Pℱ𝐾 ,𝐹𝐾,𝜎p𝐵𝜎qě0

𝐹𝐾,𝜎 p𝐵𝜎q

˛

‚𝑢𝐾 ´ÿ

𝜎Pℱ𝐾Xℱint,𝐹𝐿,𝜎p𝐵𝜎qą0

𝑢𝐿𝐹𝐿,𝜎 p𝐵𝜎q “ |𝐾|𝑓𝐾 .

Defining:𝑠𝐾 “

ÿ

𝜎Pℱ𝐾 ,𝐹𝐾,𝜎p𝐵𝜎qě0

𝐹𝐾,𝜎 p𝐵𝜎q and r𝑞𝐾 “ 𝑠𝐾𝑢𝐾

and noticing that 𝑠𝐿 ą 0 as soon as there exists 𝜎 P ℱ𝐿 such that 𝐹𝐿,𝜎 p𝐵𝜎q ą 0, we see that the above equationcan be rewritten:

r𝑞𝐾 ´ÿ

𝜎Pℱ𝐾Xℱint,𝐹𝐿,𝜎p𝐵𝜎qą0

r𝑞𝐿

𝑠𝐿𝐹𝐿,𝜎 p𝐵𝜎q “ |𝐾|𝑓𝐾 .

For any 𝜎 P ℱ𝒟ext,in, we set:

𝑠𝜎 “ ´ÿ

𝐾P𝒯𝜎,𝐹𝐾,𝜎p𝐵𝜎qă0

𝐹𝐾,𝜎 p𝐵𝜎q and r𝑞𝜎 “ 𝑠𝜎𝑢𝜎.

Gathering all those results, we obtain:ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

r𝑞𝐾 ´ÿ

𝜎Pℱ𝐾Xℱint,𝐹𝐿,𝜎p𝐵𝜎qą0

r𝑞𝐿

𝑠𝐿𝐹𝐿,𝜎 p𝐵𝜎q “ |𝐾|𝑓𝐾 @𝐾 P 𝒯

r𝑞𝜎 “ 0 @𝜎 P ℱ𝒟ext,in.

(5.2)

SOME MULTIPLE FLOW DIRECTION ALGORITHMS 1929

Notice that the above system is set for the unknowns´

pr𝑞𝐾q𝐾P𝒯 , pr𝑞𝜎q𝜎Pℱ𝒟ext,in

¯

, thus it has the same number ofunknowns than the original system (5.1). We still reconstruct a water flux vector cellwise by setting:

𝑄𝐾 “ÿ

𝜎Pℱ𝐾 ,𝐹𝐾,𝜎p𝐵𝜎qą0

r𝑞𝐾

|𝐾|𝑠𝐾𝐹𝐾,𝜎 p𝐵𝜎q p𝑥𝜎 ´ 𝑥𝐾q `

ÿ

𝜎Pℱ𝐾Xℱint,𝐹𝐾,𝜎p𝐵𝜎qă0

r𝑞𝐿

|𝐾|𝑠𝐿𝐹𝐾,𝜎 p𝐵𝜎q p𝑥𝜎 ´ 𝑥𝐾q (5.3)

the consistent water discharge being given by 𝑞𝐾 “ ||𝑄𝐾 ||. Finally, for any cell 𝐾 P 𝒯 , we set 𝑢𝐾 as we havedone for the TPFA case:

𝑢𝐾 “

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

r𝑞𝐾

𝑠𝐾if 𝑠𝐾 ‰ 0

0 otherwise(5.4)

while for any 𝜎 P ℱ𝒟ext,in, 𝑢𝜎 “ 0. Notice that this new system allows to correctly define the numerical method,while the system (5.1) does not uniquely define the water height on cells where 𝑠𝐾 “ 0. Contrary to system (3.4),system (5.2) admits no obvious renumbering of mesh elements that makes the system triangular. However, asit is anyway necessary to enable parallelism, we say that the generalized MFD algorithm will consists in solvingthe linear system (5.2) using any linear solver of the literature. Notice that as we are using the unknown r𝑞 ratherthan 𝑢, the discrete system (5.2) takes the form of a perturbation of the identity matrix. Thus, we expect itscondition number to be relatively good (depending of course of the coefficient 𝜆) and in any case much betterthan if had tried to solve (5.2) directly.

5.2. On existence and uniqueness for the generalized MFD algorithm

Existence and uniqueness of a solution to the above system are much less obvious than in the case of theTPFA scheme. To explain this fact, let us denote

𝒯 ˚ “ t𝐾 P 𝒯 | 𝑠𝐾 ą 0u .

Using the definition of 𝒯 ˚, summing (5.2) over 𝐾 P 𝒯 , after a straightforward manipulation of the last sum weget:

ÿ

𝐾P𝒯 z𝒯 ˚r𝑞𝐾 `

ÿ

𝐾P𝒯 ˚

ÿ

𝜎Pℱ𝐾 ,𝐹𝐾,𝜎p𝐵𝜎qą0

r𝑞𝐾

𝑠𝐾𝐹𝐾,𝜎 p𝐵𝜎q ´

ÿ

𝐿P𝒯 ˚

ÿ

𝜎Pℱ𝐿Xℱint,𝐹𝐿,𝜎p𝐵𝜎qą0

r𝑞𝐿

𝑠𝐿𝐹𝐿,𝜎 p𝐵𝜎q “

ÿ

𝐾P𝒯|𝐾|𝑓𝐾

and consequently:

ÿ

𝐾P𝒯 z𝒯 ˚r𝑞𝐾 `

ÿ

𝐾P𝒯 ˚

ÿ

𝜎Pℱ𝐾Xℱext,𝐹𝐾,𝜎p𝐵𝜎qą0

r𝑞𝐾

𝑠𝐾𝐹𝐾,𝜎 p𝐵𝜎q “

ÿ

𝐾P𝒯|𝐾|𝑓𝐾 . (5.5)

Thus, the existence of a solution for any second member p𝑓𝐾q𝐾P𝒯 requires that:

𝒜 “ p𝒯 z𝒯 ˚q Y 𝒯 ˚˚ ‰ H where 𝒯 ˚˚ “ t𝐾 P 𝒯 ˚ | t𝜎 P ℱ𝐾 X ℱext | 𝐹𝐾,𝜎 p𝐵𝜎q ą 0u ‰ Hu . (5.6)

In the case of the TPFA scheme, condition (5.6) is necessarily satisfied. If condition (5.6) is not satisfied,then (5.2) can have a solution only if

ÿ

𝐾P𝒯|𝐾|𝑓𝐾 “ 0.

From this observation, it seems clear that establishing an exhaustive existence and uniqueness theory for system(5.2) is a more complex task than what one could expect. For the continuous problem, well-posedness is directlylinked to the properties of the topography 𝑏 and in particular to its Laplacian. Unsurprisingly, well-posednessof (5.2) will be directly linked to the properties of the quantity that plays the role of the topography 𝑏 at the

1930 J. COATLEVEN

discrete level, that is to say the flux family p𝐹𝐾,𝜎 p𝐵𝜎qq𝐾P𝒯 ,𝜎Pℱ𝐾. Without any further assumption on the mesh,

far from the asymptotic regime consistency alone cannot be expected to ensure well-posedness for coarse meshes.This means that well-posedness will in general be controlled by the structure of the discrete flux family ratherthan by the properties of 𝑏. As in practice the Laplacian of 𝑏 cannot be computed in general, it is anyway betterto have an existence result that uses only computable quantities. In fact, not only the necessary condition (5.6)must be satisfied, but it also requires that there does not exist what we will call a flux cycle. For any subset 𝒞of 𝒯 , we denote:

ℱ𝒞int “ t𝜎 P ℱint | 𝒯𝜎 Ă 𝒞u .

We say that a set of cells 𝒞 Ă 𝒯 form a flux cycle if and only if for any 𝐾 P 𝒞

𝐹𝐾,𝜎 p𝐵𝜎q ď 0 @𝜎 P ℱ𝐾z`

ℱ𝐾 X ℱ𝒞int

˘

andÿ

𝜎Pℱ𝐾Xℱ𝒞int,𝐹𝐾,𝜎p𝐵𝜎qą0

𝐹𝐾,𝜎 p𝐵𝜎q ą 0 andÿ

𝜎Pℱ𝐾Xℱ𝒞int,𝐹𝐾,𝜎p𝐵𝜎qă0

𝐹𝐾,𝜎 p𝐵𝜎q ă 0.

The first condition implies that water can enter the cycle but not leave it, while the second condition implies thatwater will run through the cycle without stopping anywhere. Whether such configurations can exist on coarsemeshes for consistent flux families is a difficult question. However, from the physical point of view flux cyclesare completely unrealistic as they represent regions where water will at some point go from a topographicallylow region to a topographically higher one. Moreover, under the positivity hypothesis on the Laplacian of thetopography, we know that for the exact fluxes we have:

ÿ

𝜎Pℱ𝐾

ż

𝜎

´𝜆∇𝑏 ¨ 𝑛𝐾,𝜎 ą 0

and thus no flux cycle can exist. Thus, the presence of flux cycles should be considered as anomalous and asign that a too coarse mesh has been used. A natural sufficient condition of existence and uniqueness for (5.2)should then be one that is satisfied by the continuous fluxes and immediately implies that no flux cycle exist.

Proposition 5.1. We say that there exists a flowing path from 𝐾 P 𝒯 z𝒜 to 𝒜 if there exists r𝐾 P 𝒜 such that

D 𝑛𝐾 P N, 𝑛𝐾 ą 0 and p𝜎𝑖q0ď𝑖ď𝑛𝐾´1 such that 𝜎𝑖 P ℱint @0 ď 𝑖 ď 𝑛𝐾 ´ 1

and 𝒯𝜎𝑖 “ t𝐾𝑖, 𝐾𝑖`1u @0 ď 𝑖 ď 𝑛𝐾 ´ 1 where 𝐾 “ 𝐾0r𝐾 “ 𝐾𝑛𝐾

and 𝐹𝐾𝑖,𝜎𝑖p𝐵𝜎𝑖

q ą 0.

If 𝒜 ‰ H and for any cell 𝐾 P 𝒯 z𝒜 there exists a flowing path to 𝒜, then (5.2) is well-posed.

Proof. As system (5.2) is linear, it suffices to establish uniqueness to ensure the existence of solutions. Thus,assume that r𝑞 “

´

pr𝑞𝐾q𝐾P𝒯 , pr𝑞𝜎q𝜎Pℱ𝒟ext,in

¯

is solution of (5.2) with zero right-hand side. Then, taking themodulus of (5.2) and summing over 𝐾 P 𝒯 we get:

ÿ

𝐾P𝒯|r𝑞𝐾 | ď

ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾Xℱint,𝐹𝐿,𝜎p𝐵𝜎qą0

|r𝑞𝐿|

𝑠𝐿𝐹𝐿,𝜎 p𝐵𝜎q . (5.7)

Again, we have:

ÿ

𝐾P𝒯|r𝑞𝐾 | “

ÿ

𝐾P𝒯 z𝒯 ˚|r𝑞𝐾 |`

ÿ

𝐾P𝒯 ˚

ÿ

𝜎Pℱ𝐾Xℱext,𝐹𝐾,𝜎p𝐵𝜎qą0

|r𝑞𝐾 |

𝑠𝐾𝐹𝐾,𝜎 p𝐵𝜎q`

ÿ

𝐾P𝒯 ˚

ÿ

𝜎Pℱ𝐾Xℱint,𝐹𝐾,𝜎p𝐵𝜎qą0

|r𝑞𝐾 |

𝑠𝐾𝐹𝐾,𝜎 p𝐵𝜎q

while:ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾Xℱint,𝐹𝐿,𝜎p𝐵𝜎qą0

|r𝑞𝐿|

𝑠𝐿𝐹𝐿,𝜎 p𝐵𝜎q “

ÿ

𝐿P𝒯 ˚

ÿ

𝜎Pℱ𝐿Xℱint,𝐹𝐿,𝜎p𝐵𝜎qą0

|r𝑞𝐿|

𝑠𝐿𝐹𝐿,𝜎 p𝐵𝜎q .

SOME MULTIPLE FLOW DIRECTION ALGORITHMS 1931

Injecting this into (5.7), we obtain:

ÿ

𝐾P𝒯 z𝒯 ˚|r𝑞𝐾 | `

ÿ

𝐾P𝒯 ˚

ÿ

𝜎Pℱ𝐾Xℱext,𝐹𝐾,𝜎p𝐵𝜎qą0

|r𝑞𝐾 |

𝑠𝐾𝐹𝐾,𝜎 p𝐵𝜎q ď 0

which immediately implies that r𝑞𝐾 “ 0 for any 𝐾 P 𝒜. Next, let us denote 𝒯0 “ 𝒯 and 𝒜0 “ 𝒜, and define 𝒯𝑖

for any 𝑖 ě 1 by setting:𝒯𝑖 “ 𝒯 z

ď

0ď𝑘ď𝑖´1

𝒜𝑘 where 𝒜𝑖 “ p𝒯𝑖z𝒯 ˚𝑖 q Y 𝒯 ˚˚𝑖

with𝒯 ˚𝑖 “ t𝐾 P 𝒯𝑖 | 𝑠𝐾 ą 0u and 𝒯 ˚˚𝑖 “

𝐾 P 𝒯 ˚𝑖 |

𝜎 P ℱ𝐾 X ℱ 𝑖ext | 𝐹𝐾,𝜎 p𝐵𝜎q ą 0

(

‰ H(

and for any 𝑖 ě 1:

ℱ𝑖 “ t𝜎 P ℱ | 𝒯𝜎 X 𝒯𝑖 ‰ Hu and ℱ 𝑖int “ t𝜎 P ℱ𝑖 X ℱint | 𝒯𝜎 Ă 𝒯𝑖u and ℱ 𝑖

ext “ ℱ𝑖zℱ 𝑖int.

Clearly, as 𝒜 ‰ H we have 𝒯1 Ĺ 𝒯0 or 𝒯1 “ H. We now proceed by induction. Assume that for 𝑛 ě 1, we haveestablished that

r𝑞𝐾 “ 0 for all 𝐾 Pď

0ď𝑘ď𝑛´1

𝒜𝑘 and 𝒯𝑖 Ĺ 𝒯𝑖´1 or 𝒯𝑖 “ H @1 ď 𝑖 ď 𝑛´ 1.

If 𝒯𝑛 “ H, then there is nothing to prove. Otherwise, as for any 𝐾 P 𝒯𝑛, there exists a flow path to 𝒜, then𝒯 ˚˚𝑛 ‰ H. Thus, summing over 𝒯𝑛 and proceeding as above we obtain that:

ÿ

𝐾P𝒯𝑛z𝒯 ˚𝑛

|r𝑞𝐾 | `ÿ

𝐾P𝒯 ˚𝑛

ÿ

𝜎Pℱ𝐾Xℱ𝑛ext,𝐹𝐾,𝜎p𝐵𝜎qą0

|r𝑞𝐾 |

𝑠𝐾𝐹𝐾,𝜎 p𝐵𝜎q ď 0.

Thus, we obtain that r𝑞𝐾 “ 0 for any 𝐾 P 𝒜𝑛 and 𝒜𝑛 ‰ H which implies 𝒯𝑛`1 Ĺ 𝒯𝑛. Then, it is clear that thesequence p𝒯𝑖q𝑖ě0 is strictly decreasing in the sense that it satisfies 𝒯𝑖 Ĺ 𝒯𝑖´1 or 𝒯𝑖 “ H because of the flow pathassumption. Thus, there exists 𝑛𝒯 ą 0 such that 𝒯𝑛𝒯 “ H and thus r𝑞 “ 0, which concludes the proof.

6. Convergence analysis

The main idea underlying our convergence analysis is that for all consistent fluxes, if the topography 𝑏 isregular enough numerical fluxes will converge to the continuous ones. The major originality of the presentanalysis regarding for instance finite volumes or discontinuous Galerkin methods for steady-advection diffusionproblems of the form (3.2) is that here the coefficients 𝛽 and 𝜇 are approximated through discrete differentialoperators applied to the topography 𝑏. Thus, this additional approximation has to be handled carefully torecover the correct convergence estimates.

To establish precise error estimates we first need to choose a discrete norm. To this end, we define

𝜔𝐾,𝜎 “12`

|𝐹𝐾,𝜎 p𝐵𝜎q | ` |𝜎| ||𝜆∇𝑏 ¨ 𝑛𝐾,𝜎||𝐿8p𝐵p𝑥𝐾 ,2ℎ𝐾qq

˘

and set for 𝑣 “ p𝑣𝐾q𝐾P𝒯 :

||𝑣||2ℎ “𝛼𝜆´

2

ÿ

𝐾P𝒯|𝐾|𝑣2

𝐾 `12

ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾Xℱext,𝐹𝐾,𝜎p𝐵𝜎qą0

𝜔𝐾,𝜎|𝑣𝐾 |2 `

12

ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾 ,𝐹𝐾,𝜎p𝐵𝜎qă0

𝜔𝐾,𝜎p𝑣𝐾 ´ 𝑣up𝜎 q

2.

In the above expression, one would more naturally expect to find 𝐹𝐾,𝜎 p𝐵𝜎q instead of 𝜔𝐾,𝜎. However, theirbehavior is difficult to control and in particular, it is complex to estimate the way this fluxes stay away from zero

1932 J. COATLEVEN

when the continuous fluxesş

𝜎𝜆∇𝑏 ¨𝑛𝐾,𝜎 cancel creating technical difficulties, which 𝜔𝐾,𝜎 avoids. Immediately,

we get that:

|𝜔𝐾,𝜎 ´ |𝐹𝐾,𝜎 p𝐵𝜎q | | “12

ˇ

ˇ |𝐹𝐾,𝜎 p𝐵𝜎q ´ |𝜎| ||𝜆∇𝑏 ¨ 𝑛𝐾,𝜎||𝐿8p𝐵p𝑥𝐾 ,2ℎ𝐾qq

ˇ

ˇ

and thus:|𝜔𝐾,𝜎 ´ |𝐹𝐾,𝜎 p𝐵𝜎q | | ď

12𝑅𝐾,𝜎 (6.1)

where for any 𝐾 P 𝒯 and any 𝜎 P ℱ𝐾 we have denoted:

𝑅𝐾,𝜎 “ maxˆˇ

ˇ

ˇ

ˇ

𝐹𝐾,𝜎 p𝐵𝜎q `

ż

𝜎

𝜆∇𝑏 ¨ 𝑛𝐾,𝜎

ˇ

ˇ

ˇ

ˇ

,ˇ

ˇ|𝐹𝐾,𝜎 p𝐵𝜎q | ´ |𝜎| ||𝜆∇𝑏 ¨ 𝑛𝐾,𝜎||𝐿8p𝐵p𝑥𝐾 ,2ℎ𝐾qq

ˇ

ˇ

˙

. (6.2)

Notice that from (4.5), we get that there exists 𝐶 ą 0 such that for any 𝐾 P 𝒯 and any 𝜎 P ℱ𝐾 , we have:

𝑅𝐾,𝜎 ď 𝐶|𝜎|ℎ𝐾 ||𝜆||𝑊 1,8pΩq||𝑏||𝑊 2,8pΩq. (6.3)

The main objective of this section is to establish the following explicit 𝐿2 error estimate:

Theorem 6.1. Assume that the gradient operator underlying the fluxes is consistent. Further assume that themesh satisfies assumption (A1), that 𝜆 P 𝑊 1,8pΩq and that the solution 𝑢 of (3.1) belongs to 𝐻1pΩq. Let

𝛾pℎq “ max

˜

sup𝐾P𝒯 ,𝜎Pℱ𝐾 ,𝐹𝐾,𝜎p𝐵𝜎qă0

𝑅𝐾,𝜎

𝜔𝐾,𝜎, sup𝐾P𝒯 ,𝜎Pℱ𝐾Xℱext,𝐹𝐾,𝜎p𝐵𝜎q‰0

𝑅𝐾,𝜎

𝜔𝐾,𝜎

¸

with 𝑅𝐾,𝜎 defined by (6.2). Then, there exists ℐℎp𝑢q P 𝐿2pΩq such that for any 𝐾 P 𝒯 , ℐℎp𝑢q|𝐾 “ ℐ𝐾p𝑢q isconstant on 𝐾 and:

|𝑢´ ℐ𝐾p𝑢q|2𝐿2p𝐾q ` ℎ

12𝐾 |𝑢´ ℐ𝐾p𝑢q|

2𝐿2pB𝐾q ď 𝐶polyℎ𝐾 |𝑢|𝐻1p𝐾q.

Moreover if 𝛾pℎq Ñ 0 when ℎ Ñ 0, then there exists ℎ0 ą 0 and 𝐶 ą 0 depending on 𝑢, 𝜆, 𝑏 and the meshparameters such that for any ℎ ď ℎ0:

||𝑢´ ℐℎp𝑢q||ℎ ď 𝐶 maxpℎ, 𝛾pℎqq12||𝑢||𝐻1pΩq (6.4)

where 𝑢 is reconstructed by (5.4) from the solution of (5.2).

Notice that the existence of ℐℎp𝑢q in the above result is an immediate consequence of assumption (A1) and(2.3). As a corollary, we will then be able to establish an explicit error estimate for the water discharge:

Corollary 6.2. Under the assumptions of Proposition 6.1, there exists ℎ0 ą 0 and 𝐶 ą 0 depending on 𝑢, 𝜆,𝑏 and the mesh parameters such that for any ℎ ď ℎ0:

˜

ÿ

𝐾P𝒯|| ´ 𝜆𝑢∇𝑏´𝑄𝐾 ||

2𝐿2p𝐾q2

¸12

ď 𝐶 maxpℎ, 𝛾pℎqq12p1`maxpℎ, 𝛾pℎqq12q||𝑢||𝐻1pΩq. (6.5)

The error estimates use the surprising factor 𝛾pℎq instead of ℎ as one could legitimately expect. This comesfrom the fact that we use approximate fluxes, which is equivalent in some sense to using a quadrature formulafor the coefficients of (3.2). As the natural norm for the discrete problem uses those approximate fluxes, itseems unfortunately unavoidable that this impacts the error estimate. In the case where for any 𝐾 P 𝒯 and any𝜎 P ℱ𝐾 that is indeed present in the norm || ¨ ||ℎ, the 𝜔𝐾,𝜎’s are bounded by below by a constant independent onℎ, which will happen most of the time in practice, then 𝛾pℎq will behave as ℎ and we retrieve a convergence atrate ℎ12. Controlling the lower bound for 𝜔𝐾,𝜎 is the reason why we have used a supremum on a set containing

SOME MULTIPLE FLOW DIRECTION ALGORITHMS 1933

more than 𝐾 in the definition of 𝜔𝐾,𝜎. If this supremum is zero, most gradient reconstruction operators andin particular those described in our numerical experiments will provide a discrete gradient that aligns with thecontinuous one and the discrete flux will be zero too, avoiding most of the problematic cases for the asymptoticbehavior of 𝜔𝐾,𝜎. In the case of exact fluxes it is known that for the water height 𝑢 the optimal convergencerate to 𝑢 is ℎ12, even if superconvergence at rate ℎ is very often observed in practice, as revealed by [6,19] (seealso [7]). Thus, as additional flux consistency terms could only decrease the convergence rate, it seems clearthat our estimate for the water height cannot be expected to be improved in terms of order of convergence.

The proof of Theorem 6.1 being rather lengthy, we will try to decompose it as much as possible. Let us noticethat in the special case where ∇𝑏 is constant, the discrete flux are exact. Then our problem corresponds toa piecewise constant version of the discontinuous Galerkin method described in [20], and we can in principlefollow the steps of their proof. Denoting ||𝑣||2ℎ “

ř

𝐾P𝒯 ||𝑣||2𝐾,ℎ with obvious notations, the first step of their

proof is to establish a local error identity for ||𝑢𝐾 ´ 𝑣𝐾 ||𝐾,ℎ for any 𝑣𝐾 in R, using the exactness of the flux.Then, taking 𝑣𝐾 “ ℐ𝐾p𝑢q and summing over 𝐾 P 𝒯 , they obtain a global error estimate where the residualterms only involve 𝑢´ ℐp𝑢q, which can be straightforwardly estimated using polynomial approximation results.

The general guideline of our proof is the same, however we have to endure some additional technicalities dueto the non exactness of the flux. We first start by establishing the equivalent of the local error identity of [20],but keeping our approximate flux in the result. Thus, an additional residual term standing for the consistencyerror of the flux will appear on the right hand side of our identity. Then, to match the definition of the || ¨ ||ℎnorm, each time our estimates will involve the sum of the flux over the full set ℱ𝐾 , we will replace it by theLaplacian of 𝑏, and put the difference in a residual term. For an isolated flux term, we will do the same usingthis time either 𝜔𝐾,𝜎 if the term is involved in the || ¨ ||ℎ norm or by the continuous flux if it is directly a residualterm. Thus, when summing the local error identities over 𝐾 P 𝒯 to obtain the global error estimate, residualterms involving the discrete counterpart of the Laplacian will be handled through a technical result describedin the following subsection, while other residual terms will be estimated directly using the strong consistency ofthe flux. The reason for doing so is to obtain an error estimate involving only the || ¨ ||ℎ norm on the left-handside and residual terms involving either 𝑢´ ℐp𝑢q or flux consistency error terms on the right-hand side. Finally,each residual term will be estimated using polynomial approximation results and the consistency of the flux.

6.1. Local error identity and approximation results for the Laplacian

Let us now establish an abstract local error identity, following the lines of [20].

Lemma 6.3. For any 𝐾 P 𝒯 , any 𝑣𝐾 P R and any 𝜂 “ p𝜂𝜎q𝜎Pℱ𝐾P 𝐿2pB𝐾q constant on each 𝜎 P ℱ𝐾 , we have:

12

ÿ

𝜎Pℱ𝐾 ,𝐹𝐾,𝜎p𝐵𝜎qą0

𝐹𝐾,𝜎 p𝐵𝜎q p𝑢𝐾 ´ 𝑣𝐾q2`

12

ÿ

𝜎Pℱ𝐾 ,𝐹𝐾,𝜎p𝐵𝜎qă0

𝐹𝐾,𝜎 p𝐵𝜎q p𝑢up𝜎 ´ 𝜂𝜎q

2

´12

ÿ

𝜎Pℱ𝐾 ,𝐹𝐾,𝜎p𝐵𝜎qă0

𝐹𝐾,𝜎 p𝐵𝜎q pp𝑢𝐾 ´ 𝑣𝐾q ´ p𝑢up𝜎 ´ 𝜂𝜎qq

2`

12

ÿ

𝜎Pℱ𝐾

𝐹𝐾,𝜎 p𝐵𝜎q p𝑢𝐾 ´ 𝑣𝐾q2

“ÿ

𝜎Pℱ𝐾 ,𝐹𝐾,𝜎p𝐵𝜎qą0

1|𝜎|

ż

𝜎

𝐹𝐾,𝜎 p𝐵𝜎q p𝑢´ 𝑣𝐾q p𝑢𝐾 ´ 𝑣𝐾q

`ÿ

𝜎Pℱ𝐾 ,𝐹𝐾,𝜎p𝐵𝜎qă0

1|𝜎|

ż

𝜎

𝐹𝐾,𝜎 p𝐵𝜎q p𝑢´ 𝜂𝜎q p𝑢𝐾 ´ 𝑣𝐾q

´ÿ

𝜎Pℱ𝐾

ż

𝜎

ˆ

1|𝜎|

𝐹𝐾,𝜎 p𝐵𝜎q ` 𝜆∇𝑏 ¨ 𝑛𝐾,𝜎

˙

𝑢 p𝑢𝐾 ´ 𝑣𝐾q (6.6)

where 𝑢 is the solution of the continuous problem (3.1).

1934 J. COATLEVEN

Proof. This proof is a direct adaptation of its counterpart in [20]. For any 𝑤𝐾 P R and any 𝜉 “ p𝜉𝜎q𝜎Pℱ𝐾P

𝐿2pB𝐾q constant on each 𝜎 P ℱ𝐾 , consider:

ℐ𝐾 “ ´ÿ

𝜎Pℱ𝐾 ,𝐹𝐾,𝜎p𝐵𝜎qă0

𝐹𝐾,𝜎 p𝐵𝜎q p𝑤𝐾 ´ 𝜉𝜎q𝑤𝐾 `ÿ

𝜎Pℱ𝐾

𝐹𝐾,𝜎 p𝐵𝜎q𝑤2𝐾 .

Using the identity 𝑎p𝑎´ 𝑏q “ 12

`

𝑎2 ´ 𝑏2 ` p𝑎´ 𝑏q2˘

, we obtain:

ℐ𝐾 “ÿ

𝜎Pℱ𝐾

𝐹𝐾,𝜎 p𝐵𝜎q𝑤2𝐾 ´

12

ÿ

𝜎Pℱ𝐾 ,𝐹𝐾,𝜎p𝐵𝜎qă0

𝐹𝐾,𝜎 p𝐵𝜎q𝑤2𝐾 `

12

ÿ

𝜎Pℱ𝐾 ,𝐹𝐾,𝜎p𝐵𝜎qă0

𝐹𝐾,𝜎 p𝐵𝜎q 𝜉2𝜎

´12

ÿ

𝜎Pℱ𝐾 ,𝐹𝐾,𝜎p𝐵𝜎qă0

𝐹𝐾,𝜎 p𝐵𝜎q p𝑤𝐾 ´ 𝜉𝜎q2

and thus:

ℐ𝐾 “12

ÿ

𝜎Pℱ𝐾 ,𝐹𝐾,𝜎p𝐵𝜎qą0

𝐹𝐾,𝜎 p𝐵𝜎q𝑤2𝐾 `

12

ÿ

𝜎Pℱ𝐾 ,𝐹𝐾,𝜎p𝐵𝜎qă0

𝐹𝐾,𝜎 p𝐵𝜎q 𝜉2𝜎

´12

ÿ

𝜎Pℱ𝐾 ,𝐹𝐾,𝜎p𝐵𝜎qă0

𝐹𝐾,𝜎 p𝐵𝜎q p𝑤𝐾 ´ 𝜉𝜎q2`

12

ÿ

𝜎Pℱ𝐾

𝐹𝐾,𝜎 p𝐵𝜎q𝑤2𝐾 .

On the other hand, from (5.1) multiplying by 𝑤𝐾 and using the definition of 𝑢up𝜎 we have by construction:

´ÿ

𝜎Pℱ𝐾 ,𝐹𝐾,𝜎p𝐵𝜎qă0

𝐹𝐾,𝜎 p𝐵𝜎q p𝑢𝐾 ´ 𝑢up𝜎 q𝑤𝐾 `

ÿ

𝜎Pℱ𝐾

𝐹𝐾,𝜎 p𝐵𝜎q𝑢𝐾𝑤𝐾 “

ż

𝐾

𝑓𝑤𝐾 .

Let us now set 𝑤𝐾 “ 𝑢𝐾 ´ 𝑣𝐾 and 𝜉𝜎 “ 𝑢up𝜎 ´ 𝜂𝜎 for all 𝜎 P ℱ𝐾 . We obtain:

ℐ𝐾 “ ´ÿ

𝜎Pℱ𝐾 ,𝐹𝐾,𝜎p𝐵𝜎qă0

𝐹𝐾,𝜎 p𝐵𝜎q p𝜂𝜎 ´ 𝑣𝐾q𝑤𝐾 `

ż

𝐾

𝑓𝑤𝐾 ´ÿ

𝜎Pℱ𝐾

𝐹𝐾,𝜎 p𝐵𝜎q 𝑣𝐾𝑤𝐾 .

Using ´div p𝑢𝜆∇𝑏q “ 𝑓 , this leads to:

ℐ𝐾 “ ´ÿ

𝜎Pℱ𝐾 ,𝐹𝐾,𝜎p𝐵𝜎qă0

𝐹𝐾,𝜎 p𝐵𝜎q p𝜂𝜎 ´ 𝑣𝐾q𝑤𝐾 `

ż

𝐾

´div p𝑢𝜆∇𝑏q𝑤𝐾 ´ÿ

𝜎Pℱ𝐾

𝐹𝐾,𝜎 p𝐵𝜎q 𝑣𝐾𝑤𝐾 .

Integrating by parts, we get:ż

𝐾

´div p𝑢𝜆∇𝑏q𝑤𝐾 “ ´ÿ

𝜎Pℱ𝐾

ż

𝜎

𝜆∇𝑏 ¨ 𝑛𝐾,𝜎𝑢𝑤𝐾

“ÿ

𝜎Pℱ𝐾

1

|𝜎|

ż

𝜎

𝐹𝐾,𝜎 p𝐵𝜎q𝑢𝑤𝐾 ´ÿ

𝜎Pℱ𝐾

ż

𝜎

ˆ

1

|𝜎|𝐹𝐾,𝜎 p𝐵𝜎q ` 𝜆∇𝑏 ¨ 𝑛𝐾,𝜎

˙

𝑢𝑤𝐾 .

Then:

ℐ𝐾 “ ´ÿ

𝜎Pℱ𝐾 ,𝐹𝐾,𝜎p𝐵𝜎qă0

1

|𝜎|

ż

𝜎

𝐹𝐾,𝜎 p𝐵𝜎q p𝜂𝜎 ´ 𝑣𝐾q𝑤𝐾 `ÿ

𝜎Pℱ𝐾

1

|𝜎|

ż

𝜎

𝐹𝐾,𝜎 p𝐵𝜎q p𝑢´ 𝑣𝐾q𝑤𝐾

´ÿ

𝜎Pℱ𝐾

ż

𝜎

ˆ

1

|𝜎|𝐹𝐾,𝜎 p𝐵𝜎q ` 𝜆∇𝑏 ¨ 𝑛𝐾,𝜎

˙

𝑢𝑤𝐾

“ÿ

𝜎Pℱ𝐾 ,𝐹𝐾,𝜎p𝐵𝜎qă0

1

|𝜎|

ż

𝜎

𝐹𝐾,𝜎 p𝐵𝜎q p𝑢´ 𝜂𝜎q𝑤𝐾 `ÿ

𝜎Pℱ𝐾 ,𝐹𝐾,𝜎p𝐵𝜎qą0

1

|𝜎|

ż

𝜎

𝐹𝐾,𝜎 p𝐵𝜎q p𝑢´ 𝑣𝐾q𝑤𝐾

´ÿ

𝜎Pℱ𝐾

ż

𝜎

ˆ

1

|𝜎|𝐹𝐾,𝜎 p𝐵𝜎q ` 𝜆∇𝑏 ¨ 𝑛𝐾,𝜎

˙

𝑢𝑤𝐾

which finally establishes the desired identity.

SOME MULTIPLE FLOW DIRECTION ALGORITHMS 1935

In the above local error identity, one can see that the left hand side strongly looks like the 𝐾-term of the || ¨ ||ℎnorm. However, to match the || ¨ ||ℎ norm, we need to replace the sum of the fluxes over ℱ𝐾 by the Laplacianof 𝑏, as well as isolated flux by 𝜔𝐾,𝜎. If this last operation straightforwardly leads to a residual term because of(6.1), however the following lemma precise what kind of link we can expect between the discrete Laplacian andthe true Laplacian:

Lemma 6.4. For any p𝑤𝐾q𝐾P𝒯 , one has:

ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾

𝐹𝐾,𝜎 p𝐵𝜎q𝑤𝐾 “ ´ÿ

𝐾P𝒯

ż

𝐾

𝜆∆𝑏𝑤𝐾 `ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾 ,𝐹𝐾,𝜎p𝐵𝜎qă0

ż

𝜎

ˆ

1|𝜎|

𝐹𝐾,𝜎 p𝐵𝜎q ` 𝜆∇𝑏 ¨ 𝑛𝐾,𝜎

˙

ˆ p𝑤𝐾 ´ 𝑤up𝜎 q `

ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾Xℱext,𝐹𝐾,𝜎p𝐵𝜎qą0

ż

𝜎

ˆ

1|𝜎|

𝐹𝐾,𝜎 p𝐵𝜎q ` 𝜆∇𝑏 ¨ 𝑛𝐾,𝜎

˙

𝑤𝐾 .

Proof. Let us first notice that:ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾

𝐹𝐾,𝜎 p𝐵𝜎q𝑤up𝜎 “

ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾Xℱext,𝐹𝐾,𝜎p𝐵𝜎qą0

𝐹𝐾,𝜎 p𝐵𝜎q𝑤up𝜎 .

Indeed, using 𝐹𝐾,𝜎 p𝐵𝜎q ` 𝐹𝐿,𝜎 p𝐵𝜎q “ 0 for any 𝜎 P ℱint, 𝒯𝜎 “ t𝐾, 𝐿u we haveÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾Xℱint

𝐹𝐾,𝜎 p𝐵𝜎q𝑤up𝜎 “ 0

while for 𝜎 P ℱext such that 𝐹𝐾,𝜎 p𝐵𝜎q ą 0, 𝒯𝜎 “ t𝐾u, 𝑤up𝜎 “ 0 immediately implies that:

ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾Xℱext,𝐹𝐾,𝜎p𝐵𝜎qă0

𝐹𝐾,𝜎 p𝐵𝜎q𝑤up𝜎 “ 0.

Thus, we have:ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾

𝐹𝐾,𝜎 p𝐵𝜎q𝑤𝐾 “ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾 ,𝐹𝐾,𝜎p𝐵𝜎qă0

𝐹𝐾,𝜎 p𝐵𝜎q p𝑤𝐾 ´ 𝑤up𝜎 q

`ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾Xℱext,𝐹𝐾,𝜎p𝐵𝜎qą0

𝐹𝐾,𝜎 p𝐵𝜎q𝑤𝐾

using the definition of 𝑤up𝜎 . In the same way:

ÿ

𝐾P𝒯

ż

𝐾

´𝜆∆𝑏𝑤𝐾 “ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾

ż

𝜎

´𝜆∇𝑏 ¨ 𝑛𝐾,𝜎𝑤𝐾 “ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾 ,𝐹𝐾,𝜎p𝐵𝜎qă0

ż

𝜎

´𝜆∇𝑏 ¨ 𝑛𝐾,𝜎 p𝑤𝐾 ´ 𝑤up𝜎 q

`ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾Xℱext,𝐹𝐾,𝜎p𝐵𝜎qą0

ż

𝜎

´𝜆∇𝑏 ¨ 𝑛𝐾,𝜎𝑤𝐾 .

Combining those expressions gives the expected result.

6.2. Global error estimate

From the local error identity, and the link between the discrete and continuous Laplacians, we deduce a globalestimate for the discrete norm of the error:

1936 J. COATLEVEN

Lemma 6.5. Using the hypothesis and notations of Proposition 6.1, we have:

||𝑢´ ℐℎp𝑢q||2ℎ ď

ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾Xℱint,𝐹𝐾,𝜎p𝐵𝜎qă0

1|𝜎|

ż

𝜎

𝐹𝐾,𝜎 p𝐵𝜎q p𝑢´ ℐup𝜎 p𝑢qq pp𝑢

up𝜎 ´ ℐup

𝜎 p𝑢qq ´ p𝑢𝐾 ´ ℐ𝐾p𝑢qqq

`ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾Xℱext,𝐹𝐾,𝜎p𝐵𝜎qą0

1|𝜎|

ż

𝜎

𝐹𝐾,𝜎 p𝐵𝜎q p𝑢´ ℐ𝐾p𝑢qq p𝑢𝐾 ´ ℐ𝐾p𝑢qq

`ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾Xℱext,𝐹𝐾,𝜎p𝐵𝜎qă0

1|𝜎|

ż

𝜎

𝐹𝐾,𝜎 p𝐵𝜎q p𝑢´ ℐup𝜎 p𝑢qq pp𝑢𝐾 ´ ℐ𝐾p𝑢qq ´ p𝑢

up𝜎 ´ ℐup

𝜎 p𝑢qqq

´ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾

ż

𝜎

ˆ

1|𝜎|

𝐹𝐾,𝜎 p𝐵𝜎q ` 𝜆∇𝑏 ¨ 𝑛𝐾,𝜎

˙

𝑢 p𝑢𝐾 ´ ℐ𝐾p𝑢qq

´12

˜

ÿ

𝐾P𝒯

ż

𝐾

𝜆∆𝑏|𝑢𝐾 ´ ℐ𝐾p𝑢q|2 `

ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾

𝐹𝐾,𝜎 p𝐵𝜎q |𝑢𝐾 ´ ℐ𝐾p𝑢q|2

¸

´12

ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾 ,𝐹𝐾,𝜎p𝐵𝜎qă0

p|𝐹𝐾,𝜎 p𝐵𝜎q | ´ 𝜔𝐾,𝜎q pp𝑢𝐾 ´ ℐ𝐾p𝑢qq ´ p𝑢up𝜎 ´ ℐup

𝜎 p𝑢qqq2

´12

ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾Xℱext,𝐹𝐾,𝜎p𝐵𝜎qą0

p|𝐹𝐾,𝜎 p𝐵𝜎q | ´ 𝜔𝐾,𝜎q |𝑢𝐾 ´ ℐ𝐾p𝑢q|2.

Proof. We start from (6.6) with 𝑣𝐾 “ ℐ𝐾p𝑢q and 𝜂𝜎 “ ℐup𝜎 p𝑢q. Summing over 𝐾 P 𝒯 the second term of the

left hand side of (6.6) leads to:

12

ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾 ,𝐹𝐾,𝜎p𝐵𝜎qă0

𝐹𝐾,𝜎 p𝐵𝜎q p𝑢up𝜎 ´ ℐup

𝜎 p𝑢qq2

“ ´12

ÿ

𝐿P𝒯

ÿ

𝜎Pℱ𝐿Xℱint,𝐹𝐿,𝜎p𝐵𝜎qą0

𝐹𝐿,𝜎 p𝐵𝜎q p𝑢𝐿 ´ ℐ𝐿p𝑢qq2

`12

ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾Xℱext,𝐹𝐾,𝜎p𝐵𝜎qă0

𝐹𝐾,𝜎 p𝐵𝜎q p𝑢𝜎 ´ ℐ𝜎p𝑢qq2

“ ´12

ÿ

𝐿P𝒯

ÿ

𝜎Pℱ𝐿Xℱint,𝐹𝐿,𝜎p𝐵𝜎qą0

𝐹𝐿,𝜎 p𝐵𝜎q p𝑢𝐿 ´ ℐ𝐿p𝑢qq2

as both 𝑢𝜎 and ℐ𝜎p𝑢q cancels for 𝜎 P ℱ𝒟ext,in. Thus, summing over 𝐾 P 𝒯 the first two terms of the left handside of (6.6), we obtain:

12

ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾 ,𝐹𝐾,𝜎p𝐵𝜎qą0

𝐹𝐾,𝜎 p𝐵𝜎q pℎ𝑚,𝐾 ´ ℐ𝐾p𝑢qq2`

12

ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾 ,𝐹𝐾,𝜎p𝐵𝜎qă0

𝐹𝐾,𝜎 p𝐵𝜎q p𝑢up𝜎 ´ ℐup

𝜎 p𝑢qq2

“12

ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾Xℱext,𝐹𝐾,𝜎p𝐵𝜎qą0

𝐹𝐾,𝜎 p𝐵𝜎q pℎ𝑚,𝐾 ´ ℐ𝐾p𝑢qq2.

SOME MULTIPLE FLOW DIRECTION ALGORITHMS 1937

Consequently, summing over 𝐾 P 𝒯 the entire left hand side of (6.6) leads to:

ÿ

𝐾P𝒯ℐ𝐾 “

12

ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾

𝐹𝐾,𝜎 p𝐵𝜎q |𝑢𝐾 ´ ℐ𝐾p𝑢q|2

´12

ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾 ,𝐹𝐾,𝜎p𝐵𝜎qă0

𝐹𝐾,𝜎 p𝐵𝜎q pp𝑢𝐾 ´ ℐ𝐾p𝑢qq ´ p𝑢up𝜎 ´ ℐup

𝜎 p𝑢qqq2

`12

ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾Xℱext,𝐹𝐾,𝜎p𝐵𝜎qą0

𝐹𝐾,𝜎 p𝐵𝜎q |𝑢𝐾 ´ ℐ𝐾p𝑢q|2 “ 𝐹1 ` 𝐹2 ` 𝐹3.

The first term of the above expression rewrites:

𝐹1 “12

ÿ

𝐾P𝒯

ż

𝐾

´𝜆∆𝑏|𝑢𝐾 ´ ℐ𝐾p𝑢q|2

`12

˜

ÿ

𝐾P𝒯

ż

𝐾

𝜆∆𝑏|𝑢𝐾 ´ ℐ𝐾p𝑢q|2 `

ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾

𝐹𝐾,𝜎 p𝐵𝜎q |𝑢𝐾 ´ ℐ𝐾p𝑢q|2

¸

“ 𝐿0 ` 𝐿1

while the second term and third term give:

𝐹2 ` 𝐹3 “12

ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾 ,𝐹𝐾,𝜎p𝐵𝜎qă0

𝜔𝐾,𝜎 pp𝑢𝐾 ´ ℐ𝐾p𝑢qq ´ p𝑢up𝜎 ´ ℐup

𝜎 p𝑢qqq2

`12

ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾Xℱext,𝐹𝐾,𝜎p𝐵𝜎qą0

𝜔𝐾,𝜎|𝑢𝐾 ´ ℐ𝐾p𝑢q|2

`12

ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾 ,𝐹𝐾,𝜎p𝐵𝜎qă0

p|𝐹𝐾,𝜎 p𝐵𝜎q | ´ 𝜔𝐾,𝜎q pp𝑢𝐾 ´ ℐ𝐾p𝑢qq ´ p𝑢up𝜎 ´ ℐup

𝜎 p𝑢qqq2

`12

ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾Xℱext,𝐹𝐾,𝜎p𝐵𝜎qą0

p|𝐹𝐾,𝜎 p𝐵𝜎q | ´ 𝜔𝐾,𝜎q |𝑢𝐾 ´ ℐ𝐾p𝑢q|2 “ 𝐿2 ` 𝐿3 ` 𝐿4 ` 𝐿5

with obvious notations. Immediately, using the hypothesis on 𝜆 and 𝑏 we have:

𝐿0 ě𝛼𝜆´

2

ÿ

𝐾P𝒯|𝐾| p𝑢𝐾 ´ ℐ𝐾p𝑢qq

2.

Then, noticing that the second member of the sum over 𝐾 P 𝒯 of (6.6) can be decomposed into three terms𝑇1 ` 𝑇2 ` 𝑇3 with obvious notations and combining the above results, we have that

ř

𝐾P𝒯 ℐ𝐾 “ 𝑇1 ` 𝑇2 ` 𝑇3

is equivalent to ||𝑢 ´ ℐℎp𝑢q||2ℎ ď 𝑇1 ` 𝑇2 ` 𝑇3 ´ 𝐿1 ´ 𝐿4 ´ 𝐿5 as 𝐿0 ` 𝐿2 ` 𝐿3 exactly gives the square of the

|| ¨ ||ℎ norm. Using the fact that both 𝑢𝜎 and ℐ𝜎p𝑢q cancels for 𝜎 P ℱ𝒟ext,in, we obtain:

𝑇1 ` 𝑇2 “ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾Xℱint,𝐹𝐾,𝜎p𝐵𝜎qă0

1|𝜎|

ż

𝜎

𝐹𝐾,𝜎 p𝐵𝜎q p𝑢´ ℐup𝜎 p𝑢qq pp𝑢

up𝜎 ´ ℐup

𝜎 p𝑢qq ´ p𝑢𝐾 ´ ℐ𝐾p𝑢qqq

`ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾Xℱext,𝐹𝐾,𝜎p𝐵𝜎qą0

1|𝜎|

ż

𝜎

𝐹𝐾,𝜎 p𝐵𝜎q p𝑢´ ℐ𝐾p𝑢qq p𝑢𝐾 ´ ℐ𝐾p𝑢qq

`ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾Xℱext,𝐹𝐾,𝜎p𝐵𝜎qă0

1|𝜎|

ż

𝜎

𝐹𝐾,𝜎 p𝐵𝜎q p𝑢´ ℐup𝜎 p𝑢qq pp𝑢𝐾 ´ ℐ𝐾p𝑢qq ´ p𝑢

up𝜎 ´ ℐup

𝜎 p𝑢qqq

which concludes the proof.

1938 J. COATLEVEN

6.3. Estimation of the residual terms

Establishing Proposition 6.1 now just consists in estimating each term appearing in the second member ofthe above global estimate. From Lemma 6.5, we see that ||𝑢´ ℐℎp𝑢q||

2ℎ ď

ř7𝑖“1 𝑇𝑖 with obvious notations. The

first three terms account for the polynomial approximation error, while the four other terms account for theconsistency error of the flux. We now estimate those two families of residual terms separately.

Proof of Proposition 6.1: terms accounting for polynomial approximation error. Using Cauchy–Schwarzinequality and the fact that p𝑢up

𝜎 ´ ℐup𝜎 p𝑢qq ´ p𝑢𝐾 ´ ℐ𝐾p𝑢qq is constant on 𝜎 gives:

|𝑇1|2 ď

ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾Xℱint,𝐹𝐾,𝜎p𝐵𝜎qă0

ˆż

𝜎

1|𝜎||𝐹𝐾,𝜎 p𝐵𝜎q | ||𝑢´ ℐup

𝜎 p𝑢q|2

˙

||𝑢´ ℐℎp𝑢q||2ℎ.

As the gradient operator underlying the fluxes is consistent, we know from Proposition 4.5 that there exists𝐶 ą 0 such that:

ˇ

ˇ

ˇ

ˇ

1|𝜎|

𝐹𝐾,𝜎 p𝐵𝜎q

ˇ

ˇ

ˇ

ˇ

ď ||1|𝜎|

𝐹𝐾,𝜎 p𝐵𝜎q ´ 𝜆∇𝑏 ¨ 𝑛𝐾,𝜎||𝐿8p𝜎q ` ||𝜆∇𝑏 ¨ 𝑛𝐾,𝜎||𝐿8p𝜎q

ď 𝐶ℎ||𝜆||𝑊 1,8pΩq||𝑏||𝑊 2,8pΩq ` ||𝜆||𝐿8pΩq||𝑏||𝑊 2,8pΩq.

As soon as ℎ ď ℎ0 for some fixed ℎ0 this leads to the existence of some 𝐶 ą 0 independent on ℎ such that

|𝑇1|2 ď 𝐶||𝜆||𝑊 1,8pΩq||𝑏||𝑊 2,8

ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾Xℱint,𝐹𝐾,𝜎p𝐵𝜎qă0

ˆż

𝜎

|𝑢´ ℐup𝜎 p𝑢q|

2

˙

||𝑢´ ℐℎp𝑢q||2ℎ.

Then, using the definition of ℐℎp𝑢q, we getż

𝜎

|𝑢´ ℐup𝜎 p𝑢q|

2 ď 𝐶ℎ𝐾up𝜎|𝑢|2𝐻1p𝐾up

𝜎 q

where, if 𝜎 P ℱint, 𝐾up𝜎 “ 𝐾 if 𝐹𝐾,𝜎 p𝐵𝜎q ě 0 and 𝐾up

𝜎 “ 𝐿 if 𝐹𝐾,𝜎 p𝐵𝜎q ă 0, and 𝐾up𝜎 “ 𝐾

if 𝜎 P ℱext. From [7], we know that assumption (A1) implies that card pℱ𝐾q is bounded by a constant𝜌ℱ depending on the mesh parameters but independent on ℎ, and thus there exists 𝐶 ą 0 such that|𝑇1|

2 ď 2ℎ𝐶||𝜆||𝑊 1,8pΩq||𝑏||𝑊 2,8 |𝑢|2𝐻1pΩq||𝑢´ℐℎp𝑢q||2ℎ. For 𝑇2, obviously we have using Cauchy–Schwarz inequal-

ity:

|𝑇2|2 ď

ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾Xℱext,𝐹𝐾,𝜎p𝐵𝜎qą0

ˆż

𝜎

1|𝜎||𝐹𝐾,𝜎 p𝐵𝜎q | ||𝑢´ ℐ𝐾p𝑢q|

2

˙

||𝑢´ ℐℎp𝑢q||2ℎ

and thus, proceeding the same way than for 𝑇1, we obtain |𝑇2|2 ď 2ℎ𝐶||𝜆||𝑊 1,8pΩq||𝑏||𝑊 2,8 |𝑢|2𝐻1pΩq||𝑢´ℐℎp𝑢q||

2ℎ.

The situation is more delicate for 𝑇3. Indeed, ℐup𝜎 p𝑢q “ 0 for 𝜎 P ℱ𝒟ext,in, while 𝑢 “ 0 on 𝜎 P ℱext,in, thus ℐup

𝜎 p𝑢q

is not directly a good approximation of 𝑢. For any 𝜎 P ℱ𝒟ext,inXℱext,in, ℐup𝜎 p𝑢q “ 𝑢 “ 0 and thus the contribution

to 𝑇3 is zero. Then, it just remains to consider the case where 𝜎 P ℱ𝒟ext,in and 𝜎 R ℱext,in. In this case we have´ş

𝜎𝜆∇𝑏 ¨ 𝑛𝐾,𝜎 ě 0 while 𝐹𝐾,𝜎 p𝐵𝜎q ă 0. However, by definition of the residual

ˇ

ˇ𝐹𝐾,𝜎 p𝐵𝜎q `ş

𝜎𝜆∇𝑏 ¨ 𝑛𝐾,𝜎

ˇ

ˇ ď

𝑅𝐾,𝜎 which immediately implies thatˇ

ˇ

ş

𝜎𝜆∇𝑏 ¨ 𝑛𝐾,𝜎

ˇ

ˇ ď 𝑅𝐾,𝜎 and |𝐹𝐾,𝜎 p𝐵𝜎q| ď 𝑅𝐾,𝜎. Consequently, we getusing Cauchy–Schwarz inequality:

|𝑇3| ď

¨

˝

ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾Xℱext,𝐹𝐾,𝜎p𝐵𝜎qă0

1|𝜎|

𝑅𝐾,𝜎

ż

𝜎

𝑢2

˛

‚

12

ˆ

¨

˝

ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾Xℱext,𝐹𝐾,𝜎p𝐵𝜎qă0

𝑅𝐾,𝜎 pp𝑢𝐾 ´ ℐ𝐾p𝑢qq ´ p𝑢up𝜎 ´ ℐup

𝜎 p𝑢qqq2

˛

‚

12

.

SOME MULTIPLE FLOW DIRECTION ALGORITHMS 1939

Using the definition of 𝛾pℎq and the trace inequality on Ω, this leads to:

𝑇3 ď 𝛾pℎq12

¨

˝

ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾Xℱext,𝐹𝐾,𝜎p𝐵𝜎qă0

1|𝜎|

𝑅𝐾,𝜎

ż

𝜎

𝑢2

˛

‚

12

||𝑢´ℐℎp𝑢q||ℎ ď 𝐶ℎ12𝛾pℎq12||𝑢||𝐻1pΩq||𝑢´ℐℎp𝑢q||ℎ.

It finally remains to estimate the terms accounting for the consistency error on the fluxes.

Proof of Proposition 6.1: terms accounting for flux consistency error. For 𝑇4, let us first remark that:

𝑇4 “ ´ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾Xℱint

ż

𝜎

ˆ

1|𝜎|

𝐹𝐾,𝜎 p𝐵𝜎q ` 𝜆∇𝑏 ¨ 𝑛𝐾,𝜎

˙

𝑢 pp𝑢𝐾 ´ ℐ𝐾p𝑢qq ´ p𝑢up𝜎 ´ ℐup

𝜎 p𝑢qqq

´ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾Xℱext

ż

𝜎

ˆ

1|𝜎|

𝐹𝐾,𝜎 p𝐵𝜎q ` 𝜆∇𝑏 ¨ 𝑛𝐾,𝜎

˙

𝑢 p𝑢𝐾 ´ ℐ𝐾p𝑢qq .

Using the fact that both 𝑢𝜎 and ℐ𝜎p𝑢q cancel for 𝜎 P ℱ𝐾 X ℱext and 𝐹𝐾,𝜎 p𝐵𝜎q ă 0, this rewrites:

𝑇4 “ ´ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾 ,𝐹𝐾,𝜎p𝐵𝜎qă0

ż

𝜎

ˆ

1|𝜎|

𝐹𝐾,𝜎 p𝐵𝜎q ` 𝜆∇𝑏 ¨ 𝑛𝐾,𝜎

˙

𝑢 pp𝑢𝐾 ´ ℐ𝐾p𝑢qq ´ p𝑢up𝜎 ´ ℐup

𝜎 p𝑢qqq

´ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾Xℱext,𝐹𝐾,𝜎p𝐵𝜎qą0

ż

𝜎

ˆ

1|𝜎|

𝐹𝐾,𝜎 p𝐵𝜎q ` 𝜆∇𝑏 ¨ 𝑛𝐾,𝜎

˙

𝑢 p𝑢𝐾 ´ ℐ𝐾p𝑢qq “ 𝑇4,1 ` 𝑇4,2.

Cauchy–Schwarz inequality then leads to:

|𝑇4,1| ď

¨

˝

ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾 ,𝐹𝐾,𝜎p𝐵𝜎qă0

1|𝜎|

𝑅𝐾,𝜎

ż

𝜎

𝑢2

˛

‚

12

ˆ

¨

˝

ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾 ,𝐹𝐾,𝜎p𝐵𝜎qă0

𝑅𝐾,𝜎

´

p𝑢𝐾 ´ ℐ𝐾p𝑢qq ´ p𝑢up𝜎 ´ ℐup

𝜎 p𝑢qq2¯

˛

‚

12

.

Using (6.3), (2.2) and the definition of 𝛾pℎq, we easily get:

|𝑇4,1| ď 𝐶𝛾pℎq12

¨

˝

ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾 ,𝐹𝐾,𝜎p𝐵𝜎qă0

ℎ𝐾

ż

𝜎

𝑢2

˛

‚

12

||𝑢´ ℐℎp𝑢q||ℎ ď 𝐶𝛾pℎq12||𝑢||𝐻1pΩq||𝑢´ ℐℎp𝑢q||ℎ.

In the same way, Cauchy–Schwarz inequality and this time the trace inequality on Ω directly gives:

|𝑇4,2| ď 𝐶𝛾pℎq12

¨

˝

ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾Xℱext,𝐹𝐾,𝜎p𝐵𝜎qą0

ℎ𝐾

ż

𝜎

𝑢2

˛

‚

12

||𝑢´ ℐℎp𝑢q||ℎ ď 𝐶𝛾pℎq12ℎ12||𝑢||𝐻1pΩq||𝑢´ ℐℎp𝑢q||ℎ.

From Lemma 6.4, we deduce that:

|𝑇5| ďÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾 ,𝐹𝐾,𝜎p𝐵𝜎qă0

𝑅𝐾,𝜎

´

p𝑢𝐾 ´ ℐ𝐾p𝑢qq2´ p𝑢up

𝜎 ´ ℐup𝜎 p𝑢qq

2¯

`ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾Xℱext,𝐹𝐾,𝜎p𝐵𝜎qą0

𝑅𝐾,𝜎 p𝑢𝐾 ´ ℐ𝐾p𝑢qq2.

1940 J. COATLEVEN

Using p𝑎2 ´ 𝑏2q “ p𝑎´ 𝑏qp𝑎` 𝑏q and Cauchy–Schwarz inequality, we obtain:

|𝑇5| ď 𝛾pℎq12||𝑢´ ℐℎp𝑢q||ℎ

¨

˝

ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾 ,𝐹𝐾,𝜎p𝐵𝜎qă0

𝑅𝐾,𝜎 pp𝑢𝐾 ´ ℐ𝐾p𝑢qq ` p𝑢up𝜎 ´ ℐup

𝜎 p𝑢qqq2

˛

‚

12

` 𝛾pℎq12||𝑢´ ℐℎp𝑢q||ℎ

¨

˝

ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾Xℱext,𝐹𝐾,𝜎p𝐵𝜎qą0

𝑅𝐾,𝜎 p𝑢𝐾 ´ ℐ𝐾p𝑢qq2

˛

‚

12

.

Using (6.3) and assumption (A1), we immediately obtain that:ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾 ,𝐹𝐾,𝜎p𝐵𝜎qă0

𝑅𝐾,𝜎 p𝑢𝐾 ´ ℐ𝐾p𝑢qq2ď 𝐶𝜌ℱ

ÿ

𝐾P𝒯|𝐾| p𝑢𝐾 ´ ℐ𝐾p𝑢qq

2

and in the same way, using the fact that 𝑢up𝜎 “ and ℐup

𝜎 p𝑢q “ 0 on any 𝜎 P ℱext such that 𝐹𝐾,𝜎 p𝐵𝜎q ă 0 andthe commensurability of ℎ𝐾 and ℎ𝐿 if ℱ𝐾 X ℱ𝐾 ‰ H under assumption (A1):

ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾 ,𝐹𝐾,𝜎p𝐵𝜎qă0

𝑅𝐾,𝜎 p𝑢up𝜎 ´ ℐup

𝜎 p𝑢qq2ď 𝐶

ÿ

𝐿P𝒯

ÿ

𝜎Pℱ𝐿,𝐹𝐿,𝜎p𝐵𝜎qą0

|𝜎|ℎ𝐿 pℎ𝐿 ´ ℐup𝐿 p𝑢qq

2

ď 𝐶𝜌ℱÿ

𝐾P𝒯|𝐾| p𝑢𝐾 ´ ℐ𝐾p𝑢qq

2

and also:ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾Xℱext,𝐹𝐾,𝜎p𝐵𝜎qą0

𝑅𝐾,𝜎 p𝑢𝐾 ´ ℐ𝐾p𝑢qq2ď 𝐶

ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾Xℱext,𝐹𝐾,𝜎p𝐵𝜎qą0

|𝜎|ℎ𝐾 p𝑢𝐾 ´ ℐ𝐾p𝑢qq2

ď 𝐶𝜌ℱÿ

𝐾P𝒯|𝐾| p𝑢𝐾 ´ ℐ𝐾p𝑢qq

2.

Gathering those intermediate results, we get that |𝑇5| ď 𝐶𝛾pℎq12||𝑢 ´ ℐℎp𝑢q||2ℎ. Finally, we have by definition

of 𝛾pℎq and || ¨ ||ℎ that |𝑇6 ` 𝑇7| ď 𝐶𝛾pℎq12||𝑢 ´ ℐℎp𝑢q||2ℎ. Then, using the hypothesis that 𝛾pℎq goes to zero

when ℎ goes to zero, for any 𝐶 ą 0 independent on ℎ there exists ℎ0 ą 0 such that 1´𝐶 max pℎ, 𝛾pℎqq12ą 12,

and the result then immediately follows.

6.4. Error estimate for the water discharge

To establish Corollary 6.2, we will need an auxiliary discrete stability result:

Lemma 6.6. The following estimates holds:

12

ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾

𝐹𝐾,𝜎 p𝐵𝜎q𝑢2𝐾 `

12

ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾Xℱext,𝐹𝐾,𝜎p𝐵𝜎qą0

𝐹𝐾,𝜎 p𝐵𝜎q𝑢2𝐾

´12

ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾 ,𝐹𝐾,𝜎p𝐵𝜎qă0

p𝑢𝐾 ´ 𝑢up𝜎 q

2𝐹𝐾,𝜎 p𝐵𝜎q “

ÿ

𝐾P𝒯|𝐾|𝑓𝐾𝑢𝐾 . (6.7)

Proof. To establish (6.7), using the definition of 𝑢up𝜎 , we obtain multiplying (5.1) by 𝑢𝐾 and summing over

𝐾 P 𝒯 :ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾 ,𝐹𝐾,𝜎p𝐵𝜎qą0

𝐹𝐾,𝜎 p𝐵𝜎q𝑢2𝐾 `

ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾 ,𝐹𝐾,𝜎p𝐵𝜎qă0

𝐹𝐾,𝜎 p𝐵𝜎q𝑢up𝜎 𝑢𝐾 “

ÿ

𝐾P𝒯|𝐾|𝑓𝐾𝑢𝐾 .

SOME MULTIPLE FLOW DIRECTION ALGORITHMS 1941

Remark that 𝑢up𝜎 𝑢𝐾 “ 1

2

´

𝑢2𝐾 ` 𝑢𝑢𝑝2

𝜎

¯

´ 12 p𝑢𝐾 ´ 𝑢up

𝜎 q2 which immediately leads to

ÿ

𝐾P𝒯|𝐾|𝑓𝐾𝑢𝐾 “

ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾 ,𝐹𝐾,𝜎p𝐵𝜎qą0

𝐹𝐾,𝜎 p𝐵𝜎q𝑢2𝐾 `

12

ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾 ,𝐹𝐾,𝜎p𝐵𝜎qă0

𝐹𝐾,𝜎 p𝐵𝜎q𝑢2𝐾

`12

ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾 ,𝐹𝐾,𝜎p𝐵𝜎qă0

𝐹𝐾,𝜎 p𝐵𝜎q𝑢𝑢𝑝2

𝜎 ´12

ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾 ,𝐹𝐾,𝜎p𝐵𝜎qă0

𝐹𝐾,𝜎 p𝐵𝜎q p𝑢𝐾 ´ 𝑢up𝜎 q

2.

Using the fact that:ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾 ,𝐹𝐾,𝜎p𝐵𝜎qă0

𝐹𝐾,𝜎 p𝐵𝜎q𝑢𝑢𝑝2

𝜎 “ ´ÿ

𝐿P𝒯

ÿ

𝜎Pℱ𝐿Xℱint,𝐹𝐿,𝜎p𝐵𝜎qą0

𝐹𝐿,𝜎 p𝐵𝜎q𝑢2𝐿

we immediately get (6.7) as:

12

ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾 ,𝐹𝐾,𝜎p𝐵𝜎qą0

𝐹𝐾,𝜎 p𝐵𝜎q𝑢2𝐾 `

12

ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾 ,𝐹𝐾,𝜎p𝐵𝜎qă0

𝐹𝐾,𝜎 p𝐵𝜎q𝑢𝑢𝑝2

𝜎

“12

ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾Xℱext,𝐹𝐾,𝜎p𝐵𝜎qą0

𝐹𝐾,𝜎 p𝐵𝜎q𝑢2𝐾 .

Proof of Corollary 6.2. Let us denote p𝑒𝑖q0ď𝑖ď1 the canonical basis of R2. Assume that ℎ ď ℎ0, where ℎ0 isdefined in Proposition 6.1 and consider:

𝐾 “ÿ

𝜎Pℱ𝐾

𝑢𝐾

|𝐾|𝐹𝐾,𝜎 p𝐵𝜎q p𝑥𝜎 ´ 𝑥𝐾q .

We begin by establishing that˜

ÿ

𝐾P𝒯|| ´ 𝜆𝑢∇𝑏´ 𝐾 ||

2𝐿2p𝐾q2

¸12

ď 𝐶 maxpℎ, 𝛾pℎqq12||𝑢||𝐻1pΩq. (6.8)

Indeed, we have:ż

𝐾

𝜆𝐾B𝜉𝑏 “

ż

𝐾

𝜆𝐾∇𝑏∇ p𝑥𝑖 ´ 𝑥𝐾,𝑖q “ ´

ż

𝐾

𝜆𝐾∆𝑏 p𝑥𝑖 ´ 𝑥𝐾,𝑖q `ÿ

𝜎Pℱ𝐾

ż

𝜎

𝜆𝐾∇𝑏 ¨ 𝑛𝐾,𝜎 p𝑥𝑖 ´ 𝑥𝐾,𝑖q .

Using again the consistency estimate of Proposition 4.5 for the fluxes, we get:ˇ

ˇ

ˇ

ˇ

ˇ

ÿ

𝜎Pℱ𝐾

ż

𝜎

ˆ

´𝜆𝐾∇𝑏 ¨ 𝑛𝐾,𝜎 ´1|𝜎|

𝐹𝐾,𝜎 p𝐵𝜎q

˙

p𝑥𝑖 ´ 𝑥𝐾,𝑖q

ˇ

ˇ

ˇ

ˇ

ˇ

ď 𝐶|𝜎|card pℱ𝐾qℎ2𝐾 ||𝜆||𝑊 1,8pΩq||𝑏||𝑊 2,8pΩq.

And thus, using assumption (A1) we get that:ˇ

ˇ

ˇ

ˇ

´𝑢𝐾𝜆𝐾1|𝐾|

ż

𝐾

∇𝑏´ 𝐾

ˇ

ˇ

ˇ

ˇ

ď 𝐶ℎ𝐾 ||𝜆||𝑊 1,8pΩq||𝑏||𝑊 2,8pΩq|𝑢𝐾 |.

Using the triangle inequality:

|| ´ 𝜆𝑢∇𝑏´ 𝐾 ||2𝐿2p𝐾q2 ď || p𝜆´ 𝜆𝐾q𝑢∇𝑏||2𝐿2p𝐾q2 ` ||𝜆𝐾 p𝑢´ 𝑢𝐾q∇𝑏||2𝐿2p𝐾q2

` ||𝜆𝐾𝑢𝐾

ˆ

∇𝑏´1|𝐾|

ż

𝐾

∇𝑏

˙

||2𝐿2p𝐾q2 ` || ´ 𝜆𝐾𝑢𝐾1|𝐾|

ż

𝐾

∇𝑏´ 𝐾 ||2𝐿2p𝐾q2 .

1942 J. COATLEVEN

Poincare–Wirtinger’s inequality applied component by component (see [7]) gives:

||∇𝑏´1|𝐾|

ż

𝐾

∇𝑏||𝐿2p𝐾q2 ď 𝐶𝑃 ℎ𝐾 ||𝑏||𝐻2p𝐾q

where 𝐶𝑃 ą 0 is independent on ℎ under assumption (A1), and (6.8) follows from Proposition 6.1 and the aboveestimate. To conclude, it suffices to remark that by definition:

|𝐾|´

𝐾 ´𝑄𝐾

¯

“ÿ

𝜎Pℱ𝐾Xℱint,𝐹𝐾,𝜎p𝐵𝜎qă0

p𝑢𝐾 ´ 𝑢𝐿q p𝑥𝜎 ´ 𝑥𝐾q𝐹𝐾,𝜎 p𝐵𝜎q

`ÿ

𝜎Pℱ𝐾Xℱext,𝐹𝐾,𝜎p𝐵𝜎qă0

𝑢𝐾 p𝑥𝜎 ´ 𝑥𝐾q𝐹𝐾,𝜎 p𝐵𝜎q .

Thus applying Cauchy–Schwarz inequality, we get:

ÿ

𝐾P𝒯|𝐾|

ˇ

ˇ

ˇ𝐾 ´𝑄𝐾

ˇ

ˇ

ˇ

2

ď 2

¨

˝

ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾Xℱint,𝐹𝐾,𝜎p𝐵𝜎qă0

||𝑥𝜎 ´ 𝑥𝐾 ||2

|𝐾||𝐹𝐾,𝜎 p𝐵𝜎q|

`ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾Xℱext,𝐹𝐾,𝜎p𝐵𝜎qă0

||𝑥𝜎 ´ 𝑥𝐾 ||2

|𝐾||𝐹𝐾,𝜎 p𝐵𝜎q|

˛

‚||𝑢||ℎ,˚

ď 𝐶

¨

˝

ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾 ,𝐹𝐾,𝜎p𝐵𝜎qă0

|𝜎|ℎ2𝐾

|𝐾|

˛

‚ď 𝐶ℎ|Ω|||𝑢||ℎ,˚

using the bound on the flux from the proof of Proposition 6.1, and where:

||𝑢||2ℎ,˚ “12

ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾Xℱext,𝐹𝐾,𝜎p𝐵𝜎qą0

𝐹𝐾,𝜎 p𝐵𝜎q |𝑢𝐾 |2 ´

12

ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾 ,𝐹𝐾,𝜎p𝐵𝜎qă0

𝐹𝐾,𝜎 p𝐵𝜎q p𝑢𝐾 ´ 𝑢up𝜎 q

2.

From (6.7) we know that, still denoting 𝑢 the cellwise constant function of 𝐿2pΩq taking the value 𝑢𝐾 in cell 𝐾:

||𝑢||2ℎ,˚ ď ||𝑓 ||𝐿2pΩq||𝑢||𝐿2pΩq ´ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾

𝐹𝐾,𝜎 p𝐵𝜎q𝑢2𝐾 .

However, one has

ÿ

𝐾P𝒯

ÿ

𝜎Pℱ𝐾

𝐹𝐾,𝜎 p𝐵𝜎q𝑢2𝐾 “

ÿ

𝐾P𝒯

˜

ż

𝐾

∆𝑏`ÿ

𝜎Pℱ𝐾

𝐹𝐾,𝜎 p𝐵𝜎q

¸

𝑢2𝐾 ´

ÿ

𝐾P𝒯

ż

𝐾

∆𝑏𝑢2𝐾 .

Using Stokes’ formula, the consistency of the flux (4.5) and assumption (A1), we get:ˇ

ˇ

ˇ

ˇ

ˇ

ż

𝐾

∆𝑏`ÿ

𝜎Pℱ𝐾

𝐹𝐾,𝜎 p𝐵𝜎q

ˇ

ˇ

ˇ

ˇ

ˇ

“

ˇ

ˇ

ˇ

ˇ

ˇ

ÿ

𝜎Pℱ𝐾

𝐹𝐾,𝜎 p𝐵𝜎q `

ż

𝜎

𝜆∇𝑏 ¨ 𝑛𝐾,𝜎

ˇ

ˇ

ˇ

ˇ

ˇ

ď 𝐶ÿ

𝜎Pℱ𝐾

|𝜎|ℎ𝐾 ď 𝐶|𝐾|.

Thus, we have ||𝑢||2ℎ,˚ ď ||𝑓 ||𝐿2pΩq||𝑢||𝐿2pΩq ` 𝐶||𝑢||2𝐿2pΩq, and using the triangle inequality ||𝑢||𝐿2pΩq ď ||𝑢 ´

𝑢||𝐿2pΩq ` ||𝑢||𝐿2pΩq, we get that:

||𝑢||2ℎ,˚ ď 𝐶´

1`maxpℎ, 𝛾pℎqq12¯´

||𝑓 ||𝐿2pΩq `

´

1`maxpℎ, 𝛾pℎqq12¯

||𝑢||𝐻1pΩq

¯

||𝑢||𝐻1pΩq

which concludes the proof.

SOME MULTIPLE FLOW DIRECTION ALGORITHMS 1943

Figure 3. Example of meshes for the 2dDelaunay and 2dDualDelaunay mesh sequences.

7. Numerical exploration

To assess the asymptotic behavior of the method, we use a very simple analytical solution on which the positiv-ity of the Laplacian is guaranteed (contrary to typical geological applications). Let us consider the square domainΩ “s0, 𝐿rˆs0, 𝐿r associated with the topography 𝑏p𝑥, 𝑦q “ 𝑏0 ´ 𝛿

´

p𝑥´ 𝑥0q2` p𝑦 ´ 𝑦0q

2¯

where 𝑥0 “ 𝑦0 “ 𝐿2,

for which ´∆𝑏 “ 4𝛿. Consider the water height 𝑢 defined by 𝑢p𝑥, 𝑦q “ 𝑢0 exp´

´𝛼´

p𝑥´ 𝑥0q2` p𝑦 ´ 𝑦0q

2¯¯

.

Defining 𝜆p𝑥, 𝑦q “ 1, we obtain ´divp𝜆𝑢∇𝑏q “ 𝑓 with 𝑓p𝑥, 𝑦q “ 4𝛿𝑢p𝑥, 𝑦q´

1´ 𝛼´

p𝑥´ 𝑥0q2` p𝑦 ´ 𝑦0q

2¯¯

. Thecorresponding water discharge 𝑞 “ ||𝜆𝑢∇𝑏|| is given by

𝑞 “ 2𝛿𝑢p𝑥, 𝑦q´

p𝑥´ 𝑥0q2` p𝑦 ´ 𝑦0q

2¯12

.

In practice we take 𝑢0 “ 1, 𝛼 “ 2, 𝛿 “ 12, 𝑏0 “ 2 and 𝐿 “ 1. We consider seven types of mesh sequences. Thefirst sequence consists in classical Delaunay meshes (2dDelaunay). The second one (2dDualDelaunay) is obtainedby considering the dual meshes of a sequence of Delaunay meshes (Fig. 3). The third sequence (2dVoronoi) ismade of Voronoi meshes, possessing the mesh orthogonality property. The fourth sequence (2dKershawBox ) isa sequence of Kershaw meshes, while the fifth one (2dCheckerBoardBox ) is a sequence of checkerboard meshes.These two sequences have only quadrangular cells which are distorted for the sequence 2dKershawBox, whilethe sequence 2dCheckerBoardBox allows to test the behavior of the method in presence of non conformities.The sixth sequence, named 2dSquareCart, is simply a uniform cartesian mesh with square cells, while theseventh one named 2dRectCart is a uniform cartesian mesh with rectangular cells. These two last sequences aremainly intended to serve as reference. For the generalized MFD algorithm, we consider two consistent gradientreconstructions coming from finite volumes: the first one is the hybrid finite volume gradient operator of [12],that uses faces values for the topography 𝑏, while the second one is the vertex based finite volume gradientoperator (VVM) of [5].

As it is the quantity of practical interest, we will express our convergence results directly in terms of the waterdischarge 𝑞 instead of using the water height. Let us begin by some results for sequences 2dSquareCart, 2dRect-Cart and 2dVoronoi. Being the only ones that satisfy the mesh orthogonality requirement (see Figs. 4 and 5),we expect that the TPFA scheme will also converge on those sequences. The corresponding results are displayedon Figures 6 and 7, while the associated approximate convergence orders for each scheme are given in Table 1.

1944 J. COATLEVEN

Figure 4. Example of meshes for the 2dVoronoi and 2dKershawBox mesh sequences.

Figure 5. Example of meshes for the 2dCheckerBoardBox and 2dRectCart mesh sequences.

Clearly, on the basic cartesian cases 2dSquareCart and 2dRectCart, all the schemes give identical results. Acloser look at our generalized flux formula immediately reveals that this is perfectly normal as the flux 𝐹𝐾,𝜎

degenerate into the two-point flux formula on those cartesian meshes where the barycenter of each internal faceis exactly the intersection point between the face and the segment joining the cell centers on each side of theface (see [12]). Moreover, Table 1 reveals that all the methods are superconvergent on the three orthogonal meshsequences. In [20] it is indeed established in the case of exact fluxes that the approximation of the water heightsuperconverges at least on rectangular meshes, which explains that this observed superconvergence is probablynot anomalous. Next, we turn to the other mesh sequences, for which we cannot expect convergence for theTPFA scheme. The corresponding results are displayed on Figures 8 and 9, while the associated approximateconvergence orders for each scheme are given in Table 2.

As expected, the TPFA scheme does not converge anymore, however the generalized MFD algorithm isconverging for both the hybrid and virtual volume (VVM) gradients. The results for those two variants are infact relatively close to each other. In particular, the behavior of the method on sequence 2dCheckerBoardBoxconfirms its ability to deal with non conformities and thus local mesh refinement. Remark that superconvergence

SOME MULTIPLE FLOW DIRECTION ALGORITHMS 1945

Figure 6. Convergence curves for sequences 2dSquareCart and 2dRectCart.

Table 1. Approximate orders of convergence for the three schemes on orthogonal meshes.

2dSquareCart 2dRectCart 2dVoronoi

TPFA 0.943 0.942 1.147Hybrid 0.943 0.942 1.061VVM 0.943 0.942 1.061

Figure 7. Convergence curves for sequence 2dVoronoi.

1946 J. COATLEVEN

Figure 8. Convergence curves for sequences 2dDelaunay and 2dDualDelaunay.

Figure 9. Convergence curves for sequences 2dKershawBox and 2dCheckerBoardBox.

Table 2. Approximate orders of convergence for the three schemes on general meshes.

2dDelaunay 2dDualDelaunay 2dKershawBox 2dCheckerBoardBox

Hybrid 0.991 1.081 0.941 1.032VVM 0.971 1.084 1.213 1.032

SOME MULTIPLE FLOW DIRECTION ALGORITHMS 1947

Figure 10. Comparison between influx and water discharge for the TPFA scheme on orthog-onal meshes (exact water discharge is bottom right).

is also achieved again on those test cases. To conclude, on Figure 10, we represent on the bottom-right the correctwater discharge, while the top-left figure is the water influx r𝑞𝐾 for sequence 2dSquareCart, the top-right figure isthe water influx for sequence 2dRectCart and the bottom-left figure is the water influx for sequence 2dVoronoi.These very basic examples illustrate our point on the water influx r𝑞𝐾 : it is clearly a mesh dependent quantity,that cannot be expected to reproduce the behavior of the discharge correctly, even on those very simple cases. Ifone does not consider the Manning–Strickler framework, one of the tricky features of r𝑞𝐾 is that on very commoncartesian mesh sequences such as 2dSquareCart and 2dRectCart it has a relatively nice qualitative behavior.It even seems to converge, although to a mesh sequence dependent quantity. This explains why it has beenused inadvertently in the hydrogeology community, despite of its mesh dependent nature: on the widely usedcartesian meshes, it has an acceptable behavior that cannot be easily discarded without a reference continuousmodel, in particular for complex topographies. However, on more general meshes, the situation is very different,even for the TPFA scheme as the case of sequence 2dVoronoi reveals. We display on Figure 11 the water influxr𝑞𝐾 for the four other mesh sequences, in the case of the hybrid gradient. It is completely obvious on thosesequences that r𝑞𝐾 should not be considered as a physically relevant quantity.

8. Conclusion

We have established the equivalence between a classical family of multiple flow direction algorithms and theTPFA scheme applied to a family of stationary Manning–Strickler models. From this equivalence we have pro-posed an improvement of the derivation of the discrete water discharge based on a consistent flux reconstruction

1948 J. COATLEVEN

Figure 11. Water influx for hybrid gradient on meshes 2dDelaunay, 2dDualDelaunay, 2dKer-shawBox and 2dCheckerBoardBox.

rather than directly using the water influx intermediate unknown that is not only non convergent in general, butdoes not approximate the water discharge in any case. Then, using more advanced flux reconstruction schemes,we have proposed a multiple flow direction algorithm adapted to general meshes. A convergence theory thatcovers both cases was developed, and numerical experiments illustrate the good behavior of the method. Remarkthat the extension to the case of Manning–Strickler tensors rather than coefficients is straightforward, as wouldbe the extension of the present analysis to multiple flow direction models that use powers of the topographicslope.

Acknowledgements. The author would like to thank G. Enchery and L. Agelas for our numerous discussions on thecontents of this article.

References

[1] C. Bardos, Problemes aux limites pour les equations aux derivees partielles du premier ordre a coefficients reels; theoremesd’approximation; application a l’equation de transport. Ann. Sci. Ec. Norm. Sup. Ser. 4 3 (1970) 185–233.

[2] H. Beirao Da Veiga, Existence results in Sobolev spaces for a stationary transport equation. Ricerche Mat. Suppl. XXXVI(1987) 173–184.

[3] S. Brenner and R. Scott, The Mathematical Theory of Finite Element Methods, 3rd ed. Springer (2008).[4] J. Coatleven, Semi hybrid method for heterogeneous and anisotropic diffusion problems on general meshes. ESAIM: M2AN

49 (2015) 1063–1084.[5] J. Coatleven, A virtual volume method for heterogeneous and anisotropic diffusion-reaction problems on general meshes.

ESAIM: M2AN 51 (2017) 797–824.[6] B. Cockburn, B. Dong and J. Guzman, Optimal convergence of the original DG method for the transport-reaction equation

on special meshes. SIAM J. Numer. Anal. 46 (2008) 1250–1265.

SOME MULTIPLE FLOW DIRECTION ALGORITHMS 1949

[7] D.A. Di Pietro and A. Ern, Mathematical Aspects of Discontinuous Galerkin Methods. Springer (2012).

[8] R.J. DiPerna and P.L. Lions, Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98 (1989)511–547.

[9] R.H. Erskine, T.R. Green, J.A. Ramirez and L.H. MacDonald, Comparison of grid-based algorithms for computing upslopecontributing area. Water Resour. Res. 42 (2006) W09416.

[10] R. Eymard, T. Gallouet and R. Herbin, Finite volume methods, edited by P.G. Ciarlet and J.-L. Lions. In: Handbook ofNumerical Analysis: Techniques of Scientific Computing, Part III. North-Holland, Amsterdam (2000) 713–1020.

[11] R. Eymard, T. Gallouet and R. Herbin, A new finite volume scheme for anisotropic diffusion problems on general grids:convergence analysis. C. R. Math. Acad. Sci. Paris 344 (2007) 403–406.

[12] R. Eymard, T. Gallouet and R. Herbin, Discretisation of heterogeneous and anisotropic diffusion problems on general noncon-forming meshes SUSHI: a scheme using stabilisation and hybrid interfaces. IMA J. Numer. Anal. 30 (2010) 1009–1043.

[13] R. Eymard, C. Guichard and R. Herbin, Benchmark 3D: the vag scheme. In: Vol. 2 of Springer Proceedings in Mathematics,FVCA6, Prague (2011) 213–222.

[14] R. Eymard, C. Guichard and R. Herbin, Small-stencil 3D schemes for diffusive flows in porous media. ESAIM: M2AN 46(2011) 265–290.

[15] E. Fernandez-Cara, F. Guillen and R.R. Ortega, Mathematical modeling and analysis of visco-elastic fluids of the oldroyd kind,edited by P.G. Ciarlet and J.L. Lions. In: Vol. VIII of Handbook of Numerical Analysis: Numerical Methods for Fluids, Part2. North-Holland, Amsterdam (2002) 543–661.

[16] T.G. Freeman, Calculating catchment area with divergent flow based on a regular grid. Comput. Geosci. 17 (1991) 413–422.

[17] V. Girault and L. Tartar, 𝑙𝑝 and 𝑤1,𝑝 regularity of the solution of a steady transport equation. C. R. Acad. Sci. Paris, Ser. I348 (2010) 885–890.

[18] P. Holmgren, Multiple flow direction algorithms for runoff modelling in grid based elevation models: an empirical evaluation.Hydrol. Process. 8 (1994) 327–334.

[19] C. Johnson and J. Pitkaranta, An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation. Math.Comput. 46 (1986) 1–26.

[20] P. Lesaint and P.A. Raviart, On a finite element method for solving the neutron transport equation. Publ. Math. Inf. RennesS4 (1974) 1–40.

[21] C. Qin, A.-X. Zhu, T. Pei, B. Li, C. Zhou and L. Yang, An adaptive approach to selecting a flow-partition exponent for amultiple-flow-direction algorithm. Int. J. Geog. Inf. Sci. 21 (2007) 443–458.

[22] P. Quinn, K. Beven, P. Chevallier and O. Planchon, The prediction of hillslope flow paths for distributed hydrological modellingusing digital terrain models. Hydrol. Process. 5 (1991) 59–79.

[23] A. Richardson, C.N. Hill and J.T. Perron, IDA: an implicit, parallelizable method for calculating drainage area. Water Resour.Res. 50 (2013) 4110–4130.

[24] J. Seibert and B.L. McGlynn, A new triangular multiple flow direction algorithm for computing upslope areas from griddeddigital elevation models. Water Resour. Res. 43 (2007) W04501.

[25] D.G. Tarboton, A new method for the determination of flow directions and upslope areas in grid digital elevation models.Water Resour. Res. 33 (1997) 309–319.

[26] D.M. Wolock and G.J. McCabe Jr., Comparison of single and multiple flow direction algorithms for computing topographicparameters in topmodel. Water Resour. Res. 31 (1995) 1315–1324.

[27] Q. Zhou, P. Pilesjo and Y. Chen, Estimating surface flow paths on a digital elevation model using a triangular facet network.Water Resour. Res. 47 (2011) W07522.