Part 28

Mesh Adaptation / Decomposition

Singularities in lid driven cavity solved byadjusted finite element method

Pavel Burda, Jaroslav Novotny and Jakub Sıstek

Abstract The problem of singularities caused by boundary conditions is studied onthe flow in lid driven cavity. The asymptotic behaviour of pressure near the singu-larity locations is used together with the a priori error estimates of the finite elementsolution, in order to design the finite element mesh adjusted to singularity. In thisway a very precise solution in the vicinity of the singularity is obtained. A posteri-ori error estimates are used as the principal tool for error analysis. Numerical resultsshowing the dramatic improvement in pressure calculation are presented.

1 Introduction

Reliable modelling of flows in channels or tubes with abrupt changes of the di-ameter or changes in boundary conditions is still one of the challenging problems.The aim of this work is to design the finite element method (FEM) solution in thevicinity of the boundary corners precise up to chosen tolerance.

In the papers [4], [5] we studied the problem of singularities caused by non-convex corners in the flow domain. Making use of the asymptotic behaviour of thesolution near the corner and using a priori error estimates for FEM we constructedan algorithm for generating the FEM mesh. This led to very precise solution, see[4]. The precision was checked by the a posteriori error estimates.

Pavel BurdaDepartment of Mathematics, Czech Technical University, Karlovo namestı 13, CZ-121 35 Praha 2,Czech Republic, e-mail: pavel.burda@fs.cvut.cz

Jaroslav NovotnyInstitute of Thermomechanics, Academy of Sciences of the Czech Republic, Dolejskova 5, CZ-182 00 Praha 8, Czech Republic, e-mail: novotny@it.cas.cz

Jakub SıstekInstitute of Mathematics, Academy of Sciences of the Czech Republic, Zitna 25, CZ-115 67Praha 1, Czech Republic, e-mail: sistek@math.cas.cz

1

2 Pavel Burda, Jaroslav Novotny and Jakub Sıstek

In this paper similar approach is applied to the problem of lid driven cavity wherethe singularities are caused by changes of boundary conditions. We show again thatthe mesh can be designed a priori.

2 Navier-Stokes equations for incompressible viscous fluids

Let Ω be an open bounded domain in R2 filled with a fluid, and let Γ be its

boundary. The generic point of R2 is denoted by x = (x1,x2)

T considered in meters.

2.1 Steady two-dimensional flow in velocity-pressure formulation

We deal with isothermal flow of Newtonian viscous fluids with constant density.Such flow is modelled by the Navier-Stokes system of partial differential equations(nonconservative form). In this paper we deal only with steady flow:

(u ·∇)u−ν∆u+∇p = f in Ω , (1)∇ ·u = 0 in Ω , (2)

where• u = (u1,u2)

T denotes the vector of flow velocity, in m/s, being a function of x ,• p = pr

ρ is the pressure divided by the density considered in Pa m2/kg,• ν = µ

ρ denotes the kinematic viscosity of the fluid considered in m2/s,• f denotes the density of volume forces per mass unit considered in N/m2.

The system is supplied with the boundary conditionsu = g on Γ . (3)

Here g is a given function of x satisfying∫

Γ g ·n dΓ = 0, where n denotes the unitouter normal vector to the boundary Γ .

2.2 Finite element solution: a priori error estimates

For the approximate solution of the Navier-Stokes equation we use the finite ele-ment method with Taylor-Hood elements. In the paper we utilize the a priori esti-mate of the finite element error for the Navier-Stokes equations (1)-(2) (cf. [6])

‖∇(u−uh)‖L2(Ω) ≤C[(

∑K

h2kK |u|2Hk+1(TK)

)1/2+(

∑K

h2kK |p|2Hk(TK)

)1/2]

, (4)

‖p− ph‖L2(Ω) ≤C[(

∑K

h2kK |u|2Hk+1(TK)

)1/2+(

∑K

h2kK |p|2Hk(TK)

)1/2], (5)

where u, p are in turn the precise velocity vector and precise pressure, and uh, ph arein turn the approximate velocity vector and approximate pressure, hK is the diameterof triangle TK of a triangulation T , and k = 2 for Taylor-Hood elements.

Singularities in lid driven cavity 3

2.3 Steady 2D flow in stream function - vorticity form

In this section we utilize the stream function - vorticity formulation of steadyNavier-Stokes equations, which in plane geometry reads

u1∂ω∂x

+ u2∂ω∂y

= ν

(

∂ 2ω∂x2 +

∂ 2ω∂y2

)

, (6)

∂ 2ψ∂x2 +

∂ 2ψ∂y2 = −ω , (7)

u1 =∂ψ∂y

, u2 = −∂ψ∂x

, (8)

where u1,u2 are velocity components, ω is the vorticity, ψ is the stream function,and ν is the viscosity. We assume that all derivatives exist here at least in the gener-alized sense.

2.4 Asymptotic behaviour of the solution near corners

The asymptotic behaviour of plane flow with corner singularities has been studiede.g. by Kondratiev [7], Ladeveze, Peyret [8].

In this paper we investigate the plane flow in lid driven cavity, to get the asymp-totic behaviour near the upper corners of the cavity. The singularity here is causedby the sudden change of boundary condition and is qualitatively different from thatof the nonconvex corner.

Substituting (7) and (8) into (6) we get the fourth order PDE for stream func-tion. To study the asymptotic behaviour of the solution of this equation near thecorner, we first restrict ourselves (cf. [2], Kondratiev [7]) to the principal part of theequation, namely ∂ 4ψ

∂x4 +2∂ 4ψ

∂x2∂y2 +∂ 4ψ∂y4 = f , (9)

where we first take f = 0. We use the transformation into the polar coordinates ρ ,ϑ ,x = ρ cosϑ , y = ρ sinϑ . (10)

The boundary conditions for the problem of lid driven cavity are

ψ(ρ ,0) = ψ(ρ ,π2

) = 0, (11)

1ρ

∂ψ∂ϑ

(ρ ,0) = 0,1ρ

∂ψ∂ϑ

(ρ ,π2

) = 1. (12)

Now we can show that the first term of the asymptotic expansion is ρ φ(ϑ ) (cf.also Luchini [9]), where φ depends on ϑ only, i.e.

ψ(ρ ,ϑ ) = ρ φ(ϑ ) + . . . (h.o.t.). (13)Then the axial and radial components of the velocity are

uρ =1ρ

∂ψ∂ϑ

≈dφdϑ

, uϑ = −∂ψ∂ϑ

≈−ρdφdϑ

.

4 Pavel Burda, Jaroslav Novotny and Jakub Sıstek

The momentum equation (cf. e.g. Batchelor [1])

−uρ ·∂uρ

∂ρ−

1ρ

uϑ ·∂uρ

∂ϑ+

u2ϑr

+ν(∂ 2uρ

∂ρ2 +1ρ

∂uρ

∂ρ+

1ρ2

(∂ 2uρ

∂ϑ 2 −2∂uϑ∂ϑ

−uρ

))

=∂ p∂ρ

then implies

∂ p∂ρ

≈νρ2

(

φ ′′′(ϑ )+φ ′(ϑ ))

+φ 2(ϑ )

ρ+

1ρ

φ(ϑ ) ·φ ′′(ϑ ) .

After integration we get finally the asymptotics for pressurep ≈ ρ−1F(ϑ ), (14)

where the function F does not depend on ρ .Let us note that the singularity (p ≈ ρ−1) in cavity corner is more severe than

that of the corner with the angle 32 π (p ≈ ρ−0.55) investigated e.g. in [4]

3 Algorithm for generation of computational mesh

Now we combine the results of Subsections 2.2 and 2.4. By (14), the leading termof expansion for pressure is

p(ρ ,ϑ ) = ρ−1F(ϑ )+ . . . , (15)where, cf. (10), ρ is the distance from the corner, ϑ the angle and F is a smoothfunction. Taking the expansion (15), we can estimate the seminorm of p:

| p |2Hk(TK )≈C

rK∫

rK−hK

ρ2(−k−1) ρ dρ = C[

−r−2kK +(rK −hK)−2k

]

(16)

where rK is the distance of element TK from the corner, cf. Fig. 1.Putting estimate (16) into the a priori estimate (4) or (5), we derive that we should

guaranteeh2k

K

[

−r−2kK +(rK −hK)−2k

]

≈ h2kre f (17)

in order to get the error estimate of order O(hkre f ) uniformly distributed on elements.

From this expression, we compute element diameters in accordance to cho-sen hre f . We use the Newton method.

hT

rT

rT-h

T

element T

Fig. 1 Description of element variables

For evaluating the achieved accuracy of the approximate solution, we use thea posteriori error estimator, see e.g. [3].

Singularities in lid driven cavity 5

4 Numerical results

We applied the algorithm mentioned in previous section to generate the local re-finement of the finite element mesh near the upper corners of the cavity where theboundary condition for velocity changes from 1 to 0 and vice versa. Elsewhere inthe cavity we use regular rectangular mesh of 128× 128 squares. The detail of themesh near the left upper corner is in Figure 2 (right). We use this mesh to calcu-late the steady problem for lid driven cavity with Reynolds number Re = 10,000. InFigure 2 (left) there are resulting streamlines. In Figure 3 we give the comparison ofthe pressure calculated on completely uniform mesh (left) and on our locally refinedmesh (right). Figure 3 refers only to the part of the cavity shown in Figure 2 (right).The precision of the results is checked by means of a posteriori error estimates, cf.[3]. The singularity of the form (14) is thus confirmed.

X

Y

0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0 0.02 0.040.955

0.96

0.965

0.97

0.975

0.98

0.985

0.99

0.995

1

Fig. 2 Lid driven cavity, Re = 10,000, mesh 128×128 refined locally -Left: Streamlines Right: Detail of the mesh near the corner

5 Conclusion

The main achievement of this paper is that the strategy of adjusting the mesh to thecorner singularity can also be applied to the case of singularity caused by suddenchange of the boundary condition. The precision is verified by means of a posteri-ori error estimates. Also in Figure 3 we can observe dramatic improvement of thesolution when using our adjusted mesh.

Acknowledgements This work has been supported by the grant No. 106/08/0403 - GACR, by theState Research Project No. MSM 684 0770010, and by Institutional Research Plan AV0Z10190503.

6 Pavel Burda, Jaroslav Novotny and Jakub Sıstek

Fig. 3 Cavity corner -Left: Pressure on uniform mesh Right: Pressure on refined mesh

References

1. Batchelor, G., K.: An Introduction to Fluid Dynamics, Cambridge University Press (1967)2. Burda, P.: On the FEM for the Navier-Stokes equations in domains with corner singularities.

In: Krızek, M., et al. (eds.) Finite Element Methods, Supeconvergence, Post-Processing andA Posteriori Estimates, pp. 41-52. Marcel Dekker, New York (1998)

3. Burda, P., Novotny, J., Sousedık, B.: A posteriori error estimates applied to flow in a channelwith corners, Mathematics and Computers in Simulation. 61, 375–383. (2003)

4. Burda, P., Novotny, J., & Sıstek J.: Precise FEM solution of a corner singularity using anadjusted mesh. Int. J. Numer. Meth. Fluids. 47, 1285–1292. (2005)

5. Burda, P., Novotny, J., & Sıstek J.: Accurate solution of corner singularities in axisymmetricand plane flows using adjusted mesh of finite elements, in: C. Groth, D. W. Zingg. (eds.) Com-putational Fluid Dynamics 2004, Proceedings of the Third Internat. Conf. on ComputationalFluid Dynamics, ICCFD, pp. 463 - 468, Springer, Berlin (2006)

6. Girault, V., Raviart, P., G.: Finite Element Method for Navier-Stokes Equations, Springer,Berlin (1986)

7. Kondratiev, V., A.: Asimptotika resenija uravnienija Nav’je-Stoksa v okrestnosti uglovoj tockigranicy, Prikl. Mat. i Mech., 1, 119–123 (1967)

8. Ladeveze J., Peyret, R.: Calcul numerique d’une solution avec singularite des equationsde Navier-Stokes: ecoulement dans un canal avec variation brusque de section, Journal deMecanique 13, 367–396 (1974)

9. Luchini, P.: Higher-order difference approximations of the Navier-Stokes equations, Int. J.Numer. Meth. Fluids. 12, 491 – 506 (1991)

A Parallel Implementation of the BDDC Methodfor the Stokes Flow

Jakub Sıstek, Pavel Burda, Jan Mandel, Jaroslav Novotny, and Bedrich Sousedık

Abstract An application of the Balancing Domain Decomposition by Constraints(BDDC) method to the Stokes problem is presented. It is based on our parallel im-plementation that has been developed for problems of linear elasticity. We haveverified on a number of experiments for both 2D and 3D flows that the approach ispractically applicable for the Stokes problem, and thus only minor changes of thecomputational code are required. Results for a 3D extension of the problem of liddriven cavity are presented.

1 Introduction

The Balancing Domain Decomposition based on Constraints (BDDC) is one of themost advanced preconditioners for very large systems of linear equations arising

Jakub SıstekInstitute of Mathematics, Academy of Sciences of the Czech Republic,Zitna 25, Praha 1, CZ-115 67, Czech Republic, e-mail: sistek@math.cas.cz

Pavel BurdaDepartment of Mathematics, Faculty of Mechanical Engineering, Czech Technical University inPrague,Karlovo namestı 13, Praha 2, CZ-121 35, Czech Republic, e-mail: pavel.burda@fs.cvut.cz

Jan MandelDepartment of Mathematical and Statistical Sciences, University of Colorado Denver,Campus Box 170, Denver, CO 80217-3364, United States, e-mail: jan.mandel@ucdenver.edu

Jaroslav NovotnyDepartment of Mathematics, Faculty of Civil Engineering, Czech Technical University in Prague,Thakurova 7, Praha 6, CZ-166 29, Czech Republic, e-mail: novotny@mat.fsv.cvut.cz

Bedrich SousedıkDepartment of Mathematical and Statistical Sciences, University of Colorado Denver,Campus Box 170, Denver, CO 80217-3364, United States, e-mail: bedrich.sousedik@ucdenver.edu

1

2 Jakub Sıstek et al.

from the finite element method and solved by iterative substructuring. It was in-troduced by Dohrmann [2] in 2003. The underlying theory of the BDDC methodcovers problems with symmetric positive definite matrix.

The solution of the incompressible Stokes problem by a mixed finite elementmethod leads to a saddle point system with symmetric indefinite matrix. Thus, thestandard theory of BDDC does not cover this important class of problems. In thefirst attempt to apply BDDC to the incompressible Stokes problem proposed by Liand Widlund [4], the optimal preconditioning properties of BDDC were recovered.However, the approach is restricted to using discontinuous pressure approximationand requires quite nonstandard constraints between subdomains. These modifica-tions have considerably complicated the method, which has lost much of its appeal-ing simplicity.

We present a different approach. We have implemented a parallel version of theBDDC method and verified its performance on a number of problems arising fromlinear elasticity (e.g. [7]). Here, we investigate the applicability of the method andits implementation to the Stokes flow with only minor changes to the existing codefor elasticity. Although such application is beyond the standard theory of the BDDCmethod, contributive results are obtained.

Results for the Stokes flow in a three dimensional extension of the problem of liddriven cavity are presented. Mixed discretization by Taylor-Hood finite elements isused. These elements use piecewise (tri)linear pressure approximation, which doesnot allow the approach of [4], but are very popular in the computational fluid dy-namics community.

2 Stokes problem and approximation by mixed FEM

Let Ω be an open bounded domain in R2 or R

3 filled with an incompressible vis-cous fluid, and let ∂Ω be its boundary. Isothermal low speed flow of such fluid ismodelled by the following Stokes system of partial differential equations

−νΔu+ ∇p = f in Ω , (1)

−∇ ·u = 0 in Ω , (2)

u = g on ∂Ωg, (3)

−ν(∇u)n+ pn = 0 on ∂Ωh, (4)

where u denotes the vector of flow velocity, p denotes the pressure divided by the(constant) density, ν denotes the kinematic viscosity of the fluid supposed to beconstant, f denotes the density of volume forces per mass unit, ∂Ω g and ∂Ωh aretwo disjoint subsets of ∂Ω satisfying ∂Ω = ∂Ωg ∪∂Ωh, n denotes an outer normalvector to the boundary ∂Ω with unit length, and g is a given function.

We now introduce a triangulation of the domain Ω into Taylor-Hood finite ele-ments P2/P1 and/or Q2/Q1. Their application to the weak form of system (1)–(4)

A Parallel Implementation of the BDDC Method for the Stokes Flow 3

leads to the following saddle point system of algebraic equations (see e.g. [3] fordetails)

[

A BT

B 0

][

up

]

=[

f0

]

, (5)

where u denotes velocity unknowns, p denotes pressure unknowns, A and B arecalled vector–Laplacian matrix and the divergence matrix, respectively, and f is thediscrete vector of intensity of volume forces per mass unit.

If we denote A =[

A BT

B 0

]

, u =[

up

]

, and f =[

f0

]

, saddle point system (5) can

be written in the compact formAu = f . (6)

3 Iterative substructuring

Let the domain Ω be decomposed into N nonoverlapping subdomains Ω i, i =1, ...,N. Each subdomain is a union of several finite elements of the underlyingmesh, i.e. nodes of the finite elements between subdomains coincide. Unknownscommon to at least two subdomains are called the interface Γ .

Let us now write problem (5) in a refined block form, with the first block (sub-script 1) corresponding to unknowns in subdomain interiors, and the second block(subscript 2) corresponding to unknowns at the interface:

⎡

⎢

⎢

⎣

A11 A12 BT11 BT

21A21 A22 BT

12 BT22

B11 B12 0 0B21 B22 0 0

⎤

⎥

⎥

⎦

⎡

⎢

⎢

⎣

u1u2p1p2

⎤

⎥

⎥

⎦

=

⎡

⎢

⎢

⎣

f1f200

⎤

⎥

⎥

⎦

. (7)

We can define the Schur complement matrix S and the condensed right hand sideg, both with respect to interface unknowns, as

S =[

A22 BT22

B22 0

]

−[

A21 BT12

B21 0

][

A11 BT11

B11 0

]−1 [

A12 BT21

B12 0

]

,

g =[

f20

]

−[

A21 BT12

B21 0

][

A11 BT11

B11 0

]−1 [

f10

]

.

Let us denote u1 =[

u1p1

]

and u2 =[

u2p2

]

.

In iterative substructuring, problem (7) is not solved directly. Instead, unknownsat interface are first found from the reduced system

Su2 = g. (8)

4 Jakub Sıstek et al.

This is the part where the main computational effort is usually spent, and it is typi-cally done by an iterative method. At the end, interior unknowns are resolved fromthe problem

[

A11 BT11

B11 0

][

u1p1

]

=[

f10

]

−[

A12 BT21

B12 0

][

u2p2

]

. (9)

Note, that the Schur complement S is never formed explicitly and its action is real-ized by three sparse matrix multiplications and one back-substitution (cf. definitionof S). The main reason for using this approach is usually much faster convergenceof the iterative method for problem (8) compared to problem (6) (see e.g. [8]). Nev-ertheless, it can be further improved by a suitable preconditioner.

4 BDDC preconditioner

The BDDC method provides a preconditioner for problem (8). The main idea ofBDDC is to weaken most (but not all) of the constraints on continuity of finiteelement functions at interface Γ . With carefully selected set of constraints on conti-nuity among subdomains, it is cheap to solve such auxiliary problem in parallel anduse the solution for preconditioning.

More precisely, let Ai be the subdomain stiffness matrix assembled solely frommatrices of elements in subdomain i, and let Ri be the mapping of degrees of free-dom of subdomain i into all degrees of freedom. The global stiffness matrix A in (6)can be assembled from subdomain stiffness matrices as

A =N

∑i=1

RiAiRTi . (10)

In the BDDC preconditioner, matrix A is not fully assembled as in (10), but onlya partial assembly is performed at the selected coarse degrees of freedom [5]. Theseare typically unknowns at selected nodes called corners, but more general con-straints such as averages over edges or faces of subdomains should be also employedfor problems in three dimensions [6]. This partial assembly (sometimes called sub-assembly) corresponds to matrix denoted ˜A. It is larger than the original matrix of theproblem A, but it possesses a simpler structure suitable for direct solution methods.This is the reason why it can be used for preconditioning.

While matrix ˜A is simpler for inverting, it has larger dimension than A – most ofunknowns at interface are contained twice, some of them even more times. To pro-duce the preconditioned residual in the space corresponding to the original problem(6), the projection E is defined. It is realized as a weighted average of values fromdifferent subdomains at unknowns on the interface Γ , thus resulting in functionscontinuous across the interface.

Let r be the residual in an iteration of an iterative method extended from interfaceto interiors of subdomains by zeros. The BDDC preconditioner MBDDC producesthe preconditioned residual v as MBDDC : r → v = Ew, where w is obtained as the

A Parallel Implementation of the BDDC Method for the Stokes Flow 5

solution to problem˜Aw = ET r. (11)

After v is found, we are typically interested only in its values at interface nodes,since multiplication of this vector by S follows.

5 Implementation and numerical results

Our parallel implementation of BDDC is based on the multifrontal massively par-allel sparse direct solver MUMPS [1] (version 4.8.4), which is used for the factor-ization of matrix ˜A in (11) and matrix of problem (9). These matrices are put intoMUMPS in the distributed format with one subdomain corresponding to one proces-sor. For the Stokes problem, MUMPS has been used for the LDLT factorization ofgeneral symmetric matrices. The iterations are performed by a parallel PCG solver.

The method is tested on the problem of lid driven cavity. This popular 2D bench-mark problem is used in the following 3D setting: The domain is a unit cube withhomogeneous Dirichlet boundary conditions on all faces except unit tangential ve-locity prescribed on the upper side (called lid). The velocity is rotated by angle π/8.The entire motion inside cavity is driven by viscosity of the fluid which is chosento 0.01. Since Dirichlet boundary condition for velocity is prescribed on the entireboundary of the domain (sometimes called enclosed flow), we fix pressure at a nodein the centre of the domain to make the pressure solution unique.

The mesh consists of 32×32×32 = 32,768 Taylor-Hood finite elements and457,380 unknowns, and it was divided into 32 irregular subdomains by METISgraph partitioner (Fig. 1 left). The stopping criterion was chosen as ‖r‖ 2/‖g‖2 <10−6. The problem requires 46 PCG iterations and takes 731 seconds on 32 proces-sors of SGI Altix 4700 computer. In Fig. 1 (right), the streamline through point withcoordinates [0.5,0.55,0.5] is plotted.

6 Conclusion

In our contribution, we present a parallel implementation of the BDDC precon-ditioner. After a verification of the solver on a number of problems from linearelasticity analysis, we explore the application of BDDC to problems with indefi-nite matrices, namely the Stokes problem. Although the available theory either doesnot cover this case, or treats it differently [4, 9], our experiments suggest promisingways for this effort. Results for a benchmark problem of the lid driven cavity in3D are presented. These results show that the BDDC preconditioner in its originalform is practically applicable to the Stokes flow and may considerably speed up thesolution.

6 Jakub Sıstek et al.

Fig. 1 3D lid driven cavity problem; division into 32 irregular subdomains (left); the streamlinethrough point with coordinates [0.5,0.55,0.5], viscosity 0.01 (right).

Acknowledgements

This research has been supported by the Czech Science Foundation under Grant GACR 106/08/0403, by National Science Foundation under Grant DMS-0713876, andby projects MSM 6840770001 and MSM 6840770010. It has been also supportedby Institutional Research Plan AV0Z10190503.

References

1. Amestoy, P.R., Duff, I.S., L’Excellent, J.Y.: Multifrontal parallel distributed symmetric andunsymmetric solvers. Comput. Methods Appl. Mech. Engrg. 184, 501–520 (2000)

2. Dohrmann, C.R.: A preconditioner for substructuring based on constrained energy minimiza-tion. SIAM J. Sci. Comput. 25(1), 246–258 (2003). DOI 10.1137/S1064827502412887

3. Elman, H.C., Silvester, D.J., Wathen, A.J.: Finite elements and fast iterative solvers: with appli-cations in incompressible fluid dynamics. Numerical Mathematics and Scientific Computation.Oxford University Press, New York (2005)

4. Li, J., Widlund, O.: BDDC algorithms for incompressible Stokes equations. SIAM J. Numer.Anal. 44(6), 2432–2455 (2006). DOI 10.1137/050628556

5. Li, J., Widlund, O.B.: FETI-DP, BDDC, and block Cholesky methods. Internat. J. Numer.Methods Engrg. 66(2), 250–271 (2006)

6. Mandel, J., Dohrmann, C.R.: Convergence of a balancing domain decomposition by constraintsand energy minimization. Numer. Linear Algebra Appl. 10(7), 639–659 (2003)

7. Sıstek, J., Novotny, J., Mandel, J., Certıkova, M., Burda, P.: BDDC by a frontal solver andstress computation in a hip joint replacement. Math. Comput. Simulation 80(6), 1310–1323(2010). DOI 10.1016/j.matcom.2009.01.002

8. Toselli, A., Widlund, O.: Domain Decomposition Methods—Algorithms and Theory, SpringerSeries in Computational Mathematics, vol. 34. Springer-Verlag, Berlin (2005)

9. Tu, X.: A BDDC algorithm for mixed formulation of flow in porous media. Electron. Trans.Numer. Anal. 20, 164–179 (2005)

An Adaptive Multiwavelet-based DGDiscretization for Compressible Fluid Flow

Francesca Iacono, Georg May, Siegfried Muller and Roland Schafer

1 Introduction

Multiresolution-based mesh adaptation using biorthogonal wavelets has been quitesuccessful with Finite Volume (FV) solvers for compressible fluid flow (see [2]).The rationale behind its design is to accelerate a given FV scheme (referencescheme) on a uniformly refined mesh (reference mesh) through computing actuallyonly on a locally refined adapted subgrid, while preserving (up to a fixed constantmultiple) the accuracy of the discretization on the full uniform grid. For this pur-pose, a multiresolution analysis (see [5]) is performed using biorthogonal wavelets,i.e., the array of cell averages associated with any given FV discretisation is trans-formed into a different format that reveals insight into the local behaviour of thesolution. This format allows for data compression in regions where the solutionis locally smooth. From the compressed data set a locally refined grid can be de-termined. Hence, the grid becomes sparser with higher compression rates that arerelated to an increasing number of vanishing moments of the biorthogonal wavelets.

However, for biorthogonal wavelets to realize more vanishing moments, extend-ing the support of the wavelet functions is required. The construction of such sup-port, in particular, becomes even more complicated on unstructured grid hierar-chies. On the other hand, increasing the order of accuracy for FV schemes alsorequires larger stencils. To overcome this bottleneck, Discontinuous Galerkin (DG)discretizations have become quite popular. Here, higher order can be achieved byincreasing the number of polynomial scaling functions per mesh element, i.e., inaddition to the cell average there are also higher-order coefficients.

Francesca Iacono, Georg MayAICES, RWTH Aachen, Schinkelstrasse 2, 52056, Aachen, Germany, e-mail: iacono@aices.rwth-aachen.de, may@aices.rwth-aachen.de

Siegfried Muller, Roland SchaferIGPM, RWTH Aachen, Templergraben 55, 52056, Aachen, Germany, e-mail: mueller@igpm.rwth-aachen.de, schaefer@igpm.rwth-aachen.de

1

2 Francesca Iacono, Georg May, Siegfried Muller and Roland Schafer

The extension of the multiresolution-based mesh adaptation concept to higher-order DG discretizations may be done using so-called multiwavelets ([6]). Multi-wavelets allow for higher-order vanishing moments, while being defined on onemesh element. An implementation for scalar one-dimensional conservation lawshas been developed and tested ([4]).

Here we provide a proof of concept for a solution methodology for compressibleflow, using a multiwavelet-based DG discretization. A suitable adaptation algorithmbased on the multiwavelet framework is established in Section 2. In Section 3 wethen verify the concept by means of a steady state quasi-1D nozzle flow.

2 Multiscale-Based Spatial Grid Adaptation

In order to improve the efficiency of the reference DG scheme we use multiscale-based grid adaptation techniques. Here the refinement criterion is not based on aposteriori error estimates, typically not available for compressible flow equations.Instead, we perform a multiscale analysis of the discrete data and apply thresholdtechniques to compress the data. In the following we briefly summarize the basicconceptual ideas. For technical details we refer the reader to [4].

Step 1: Multiscale analysis. Let m be the degree of the polynomials. Let uL =uL,k,ik∈IL, i=0,...,m be the array of single-scale coefficients representing the dis-cretized flow field at some fixed time level tn on a given uniform highest levelof resolution l = L (reference mesh) associated with a given DG discretization(reference scheme). Then we perform a multiscale decomposition associated witha hierarchy of nested grids Gl with increasing resolution l = 0, . . . ,L determinedby dyadic grid refinement. In Figure 1 a dyadic grid hierarchy Gl = k 2−lk∈Ilwith Il = k = 0, . . . ,2l on the domain Ω = [0,1] is illustrated. The fundamen-tal idea is to rewrite the array of single-scale coefficients as a sequence of single-scale coefficients representing an approximate solution on some coarsest levell = 0, where the fine scale information is encoded in arrays of detail coefficientsdl = dl,k,ik∈Il , i=0,...,m, l = 0, . . . ,L−1 of ascending resolution, see Figure 2.

Fig. 1 Sequence of nested grids. Fig. 2 Multiscale transformation.

Step 2: Thresholding. It can be shown that the detail coefficients become smallwith increasing refinement level when the underlying function is smooth. This isillustrated in Figure 3 in case of cell averages corresponding to a first order DGscheme, where the local two-scale decomposition is realized by the Haar wavelet. Inorder to compress the original data this motivates us to discard all detail coefficients

An Adaptive Multiwavelet-based DG Discretization for Compressible Fluid Flow 3

dl,k whose absolute values fall below a level-dependent threshold value εl = 2l−Lε .Let DL,ε be the set of significant details. The ideal strategy would be to determinethe threshold value ε such that the discretization error of the reference scheme, i.e.,difference between exact solution and reference scheme, and the perturbation error,i.e., the difference between the reference DG scheme and the adaptive scheme, arebalanced, see [4].

Fig. 3 Illustration of local two-scale decompo-sition in case of cell averages

Fig. 4 Top: Data; Bottom: Cell centers of adap-tive grid associated to the local refinement level.

Step 3: Prediction. Since the flow field evolves in time, grid adaptation is per-formed after each evolution step to provide the adaptive grid at the new time level.In order to guarantee the adaptive scheme to be reliable in the sense that no sig-nificant future feature of the solution is missed, we have to predict all significantdetails at the new time level n + 1 by means of the details at the old time level n.Let Dn+1

L,ε ⊃DnL,ε ∪Dn+1

L,ε be the prediction set. For the prediction strategy we applyHarten’s heuristic idea as is detailed in [5]. Note that prediction is now applied notonly to the details corresponding to the cell averages, but also to those correspondingto the higher order coefficients of the DG scheme.

Step 4: Grid adaptation. By means of the set Dn+1L,ε a locally refined grid is deter-

mined. For this purpose, we recursively check, proceeding levelwise from coarse tofine, whether there exists a significant detail to a cell. If there is one, then we refinethe respective cell. We finally obtain the locally refined grid with hanging nodesrepresented by the index set GL,ε . An example for data exhibiting a jump is shownin Figure 4. It is then on the adaptive grid that we perform time evolution.

3 Numerical Results

Let us consider isentropic, quasi-1D nozzle flow (see [1]). Its governing equationsare

∂w(t,x)∂ t

+∂ f(w(t,x))

∂x= s(x,w(t,x)), (t,x) ∈ (0,T )× [0,1], (1)

4 Francesca Iacono, Georg May, Siegfried Muller and Roland Schafer

where

w =

ρ

ρuρE

, f =

ρuρu2 + p

u(ρE + p)

, s =− 1A

dAdx

ρuρu2

u(ρE + p)

. (2)

We define a cross section of the nozzle on the interval [xin,xout ] = [0,1] by

A(x) =

din +(ξ (x))2 (3−2ξ (x))(dthr −din), x < xthr,dthr +(η(x))2 (3−2η(x))(dout −dthr), x ≤ xthr,

(3)

where ξ (x) =x

xthr, and η(x) =

x− xthr

xout − xthr. The throat of the nozzle is located at

xthr = 0.375, whereas the diameter of the nozzle at the inlet, throat, and outlet isgiven by din = 1, dthr = 0.875, and dout = 1.25, respectively. The in- and outfloware chosen subsonic. We prescribe the values of the entropy and enthalpy at the inlet,and the value of the pressure at the outlet.

3.1 Reference DG Scheme

In our reference DG scheme, Roe’s numerical flux, Shu’s limiter and 3-stage TVD-Runge-Kutta scheme by Shu and Osher are used, as in [3]. Moreover, polynomialsof degree m = 3 are chosen for the 4th order accurate DG discretization. In orderto check that the nominal order of accuracy is recovered, we choose a test case ex-hibiting a shock-free and, hence, smooth solution. By choosing boundary conditionsso that Mach number is M = 0.3 at the nozzle exit, the flow is subsonic inside thewhole nozzle (see Figure 5).

Fig. 5 Mach number for the isentropic quasi-1Dnozzle flow.

Fig. 6 Mach number for the quasi-1D nozzleflow with supersonic flow in a section of the noz-zle.

Table 1 reports the steady-state entropy production in mesh refinement for the4th order DG scheme. If the flow, like in this case, is shock-free, the solution shouldapproach ∆s = 0, where

An Adaptive Multiwavelet-based DG Discretization for Compressible Fluid Flow 5

∆s =p

p∞

(ρ∞

ρ

)γ

−1. (4)

is the entropy production defined as deviation from the free-stream entropy. We canobserve that the DG scheme converges at the correct rate m + 1 until machine zerois approached.

Table 1 4th order DG scheme: Convergence in mesh refinement for isentropic quasi-1D nozzleflow. The study is performed for the discrete `∞ norm and the weighted discrete `1 norm.

Cells ‖∆s‖∞ Order 1n‖∆s‖1 Order

10 1.679542e-06 2.729009e-0730 3.937016e-08 3.416372 3.365142e-09 4.00108350 4.661228e-09 4.177021 4.746640e-10 3.834221

100 3.269503e-10 3.833567 2.992431e-11 3.987517200 2.160316e-11 3.919757 1.859716e-12 4.008164

3.2 Adaptive DG Scheme

In order to test the adaptive feature of our DG method, we also consider a testcase exhibiting a shock, i.e., a discontinuity in the solution. We choose boundaryconditions so that supersonic flow develops in an internal section of the nozzle.Starting from subsonic Mach number at the inlet, the flow gradually reaches thesupersonic regime. At about x = 0.644 a shock occurs and the flow becomes againsubsonic (see Figure 6).

Limiting is needed to capture shocks and discontinuities which arise in hyper-bolic equations. Multiscale analysis supports limiting strategies in a way tightlyrelated to mesh adaptation. In regions of very steep gradients, or shocks, mesh re-finement in conjunction with lower-order approximation is reasonable. This is rel-atively easy to implement, provided one may safely assume that cells on the finestlevel of discretization are in regions where the treatment with low-order approxi-mation becomes necessary. This assumption is underpinned by the multiresolutionanalysis, which acts as a shock detector. Such an approach ensures that limiters arenot active in smooth regions, such as non-sonic critical points.

We fix the threshold value ε to be 0.01 and we apply our approach to the testcases introduced above. Results are reported in Figures 7 and 8. The adaptive so-lutions show no visible difference from the exact solutions. For each level we alsorepresent the cells which are part of the adaptive grid in correspondence of theirspatial position. For each cell Vk on level l, the absolute value of the 4 details |dl,k,i|,i = 0, . . . ,3 (from bottom to top) is represented by means of a grey scale (different

6 Francesca Iacono, Georg May, Siegfried Muller and Roland Schafer

Fig. 7 Subsonic isoentropic quasi-1D nozzleflow: solution at steady state with the adap-tive DG procedure. Top: Mach number. Bottom:adaptive grid after hard thresholding.

Fig. 8 Chocked quasi-1D nozzle flow: solutionat steady state with the adaptive DG procedure.Top: Mach number. Bottom: adaptive grid afterhard thresholding.

on each level l), where white corresponds to 0 and black to the maximal value onthat level. The magnitude of the scales is given as label on the right axis.

We can observe that the details are small and, hence, no grid refinement is trig-gered when the solution is continuous, while large details appear and the grid isrefined up to the highest level L = 10 in correspondence of the shock. However, weare not just interested in preserving the accuracy of the reference DG scheme, butwe also want to accelerate the DG scheme. In particular, the adaptive grid containsless cells, i.e., less DOFs which have to be evolved in time. The number of cells inthe adaptive grid after the prediction step is about 0.19% of the number cells in thereference DG scheme on level L = 10 for the first test case, and 0.35% for the secondtest case. This corresponds to a compression rate of about 526 and 286, respectively.

Acknowledgements Financial support from the Deutsche Forschungsgemeinschaft (German Re-search Association) through grant GSC 111 is gratefully acknowledged.

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