numerical performance of triangle element approximation ... · 4-edgaor method was implemented by...

International Journal of Mathematical AnalysisVol. 9, 2015, no. 54, 2667 - 2679HIKARI Ltd, www.m-hikari.com

http://dx.doi.org/10.12988/ijma.2015.59231

Numerical Performance of Triangle Element

Approximation for Solving 2D Poisson Equations

Using 4-Point EDGAOR Method

Mohd Kamalrulzaman Md Akhir, Jumat Sulaiman

Faculty of Science and Natural ResourcesUniversiti Malaysia Sabah

88400 Kota Kinabalu Sabah Malaysia

Copyright c© 2015 Mohd Kamalrulzaman Md Akhir and Jumat Sulaiman. This article isdistributed under the Creative Commons Attribution License, which permits unrestricted

use, distribution, and reproduction in any medium, provided the original work is properly

cited.

Abstract

This paper aims to examine the implementation of the Explicit Decou-pled Group (EDG) methods. More specifically, the four-point ExplicitDecoupled Group Accelerated Over-relaxation (4-EDGAOR) is investi-gated. Owing to the efficacy of this method, the main aim of the currentresearch is to demonstrate the benefits of the 4-EDGAOR in solvingtwo-dimensional (2D) Poisson equations with the help of the half-sweeptriangle finite element approximation equation based on the Galerkinscheme. The results of numerical experiments demonstrate the effec-tiveness of the 4-EDGAOR method over the previous four point blockmethods (4-EDGSOR, 4-EDG, 4-EGAOR, 4-EGSOR and 4-EGGS).Based on the numerical results obtained, the results show that the 4-EDGAOR method outperforms the 4-EDGSOR, 4-EDG, 4-EGAOR, 4-EGSOR and 4-EG methods in terms of number of iterations and CPUtime.

Mathematics Subject Classification: 41A55, 45A05, 45B05

Keywords: Partial Differential equations (PDEs); Poisson equation, Ex-plicit Decoupled Group (EDG), Point Block Iteration; Galerkin Scheme, Tri-angle element, AOR method

2668 Mohd Kamalrulzaman Md Akhir and Jumat Sulaiman

1 Introduction

Partial Differential equations (PDEs) are implemented in mathematical modelsin numerous and diverse physical circumstances, as well as in re-formulationsof other mathematical problems. As evident in the related literature, PDEsof the Poisson type are among the most practical and frequently investigated.Therefore, in this paper, numerical solutions of linear, 2D Poisson equationsare considered. The standard form for 2D Poisson equations can be representedmathematically as follows:

∂2U

∂x2+

∂2U

∂y2= f (x, y) , (x, y) ∈ [a, b]× [a, b] (1)

with the dirichlet boundary conditions

U(x, a) = g1(x), a ≤ x ≤ b, U(x, b) = g2(x), a ≤ x ≤ b,U(a, y) = g3(y), a ≤ x ≤ b, U(b, y) = g4(x), a ≤ y ≤ b,

where f (x, y) is a given function with sufficient smoothness. A numerical ap-proach to the solution of the problem (1) is a fundamental subdivision of sci-entific examination. Basically, 2D Poisson equations are solved numerically bydiscretizing the problems to the solution of linear systems. In order to obtainnumerical solutions, some valid numerical methods for discretizing the prob-lem (1) have been developed in recent years, including the mesh-free [4, 20, 21]and mesh-based methods. However, these discretization schemes mostly leadto large sparse linear systems that could involve high price in order to providethe solution, based on direct methods, with the escalation in the order of thelinear systems. Hence, a substitute of these schemes could be the iterativemethods that provide proficient solutions for the above-mentioned extensiveproblems

Amongst the existing iterative methods, point block methods have been ex-tensively accepted as efficient methods for sparse linear systems. The standardblock method (also known as the 4-EGGS method [6]) is a particular exampleof the four point block method. Apart from the standard 4-EGGS method,the variants of the four point block method, which are 4-EDGSOR methods,have also been proposed by [1]. Additional research studies were executed by[9, 10, 12], to authenticate the efficacy of the half-sweep concept. Fundamen-tally, the EDG method is derived from a complexity reduction approach basedon half- sweep concepts, respectively. To introduce the 4-EDGAOR methodbased on the Galerkin scheme in solving 2D Poisson equations, is the foremostobjective of current research paper.

As shown in Figure 1, in order to simplify the formulation of the full-sweepand half-sweep triangle element approximation equations for problem (1), uni-

Numerical performance of triangle element approximation 2669

(a) (b)

Figure 1: (a) and (b) show the solution domain Ω of triangle elements for thefull- and half-sweep cases at n = 8.

form node points are discussed only. Based on Figure 1, there is a need todiscretize the solution of domain evenly in both the x and y directions with amesh size h, as given below:

∆x = ∆y = h =b− an

, m = n+ 1. (2)

As depicted in Figure 1, for problem (1), the triangle finite element net-works were built as a guide to derive triangle finite element approximationequations. Correspondingly, the similar concept of the half-sweep iterationswas implemented in the finite difference methods [1, 16]; each triangle elementinvolved only two node points of type, as revealed in Figure 1. Thus, thefull-sweep and half-sweep approaches were implemented in the same type •of node points until the iterative convergence test was achieved. Afterwards,at the remaining points (points of type ◦), other approximate solutions werecomputed directly [1, 13, 16].

The framework of this paper is ordered in the sequence explained subse-quently. Starting with Section 2, derivation of the half-sweep triangle elementapproximation is elaborated at first. Followed by the Section 3 that discussesthe implementation of the 4-EDGAOR method for solving problems (1). Sub-sequently, in Section 4, numerical results are shown to assess the performanceof the examined iterative methods followed by the concluding remarks that arepresented in Section 5.


2 Formulation of Half-sweep Triangle Element

Approximations

As discussed in the aforementioned section, to solve 2D Poisson equations, the4-EDGAOR method was implemented by using the half-sweep finite elementapproximation equation based on the Galerkin scheme. The common approx-imation of the function U(x, y), by keeping in view only three node points oftype, in the form of an interpolation function for a haphazard triangle element,e, is given by [3, 4]:

Ũ [e](x, y) = N1(x, y)U1 +N2(x, y)U2 +N3(x, y)U3 (3)

The shape functions Nk(x, y), k = 1, 2, 3, can generally be shown as:

Nk (x, y) =1

|A|(ak + bkx+ cky) , k = 1, 2, 3 (4)

where,

|A| = x1 (y2 − y3) + x2 (y3 − y1) + x3 (y1 − y2)

a1a2a3

= x2y3 − x3y2x3y1 − x1y3x1y2 − x2y1

, b1b2b3

= y2 − y3y3 − y1y1 − y2

, c1c2c3

= x3 − x2x1 − x3x2 − x3

,Beside this, the rst order partial derivatives of the shape functions towards xand y can be shown, respectively, as follows:

∂∂x

(Nk (x, y)) =bk

detA∂∂y

(Nk (x, y)) =ck

detA

}, k = 1, 2, 3 (5)

According to Figure 2, using the approximation of the functions U(x, y) andf(x, y), the distribution of the hat function, Rr,s (x, y), in the solution domain,in the case of the full- and half-sweep for the whole domain can be defined asgiven below and was also described in [19]:

Ũ(x, y) =m∑r=0

m∑s=0

Rr,s(x, y)Ur,s (6)

f̃(x, y) =m∑r=0

m∑s=0

Rr,s(x, y)Ur,s (7)


(a) (b)

Figure 2: (a) and (b) show the definition of the hat function Ri,j(x, y), of full-and half-sweep triangle elements at the solution domain.

and

Ũ(x, y) =m∑

r=0,2,4

m∑s=0,2,4

Rr,s(x, y)Ur,s +m−1∑r=1,3,5

m−1∑s=1,3,5

Rr,s(x, y)Ur,s (8)

f̃(x, y) =m∑

r=0,2,4

m∑s=0,2,4

Rr,s(x, y)Ur,s +m−1∑r=1,3,5

m−1∑s=1,3,5

Rr,s(x, y)Ur,s (9)

Hence, Eq. (8) is is an approximate solution of the problem (1). For problem(1), in order to develop the full-sweep and half-sweep finite element approxi-mation equations, Galerkin scheme [18] was considered as follows:∫∫

D

Ri,j(x, y)Ei,j(x, y) = 0, i, j = 0, 1, 2, . . . ,m (10)

where, E(x, y) = ∂2U∂x2

+ ∂2U∂y2− f(x, y) is a residual function. By incorporating

the Green theorem, Eq. (10) can be rewritten as:

∫λ

(−Ri,j(x, y)

∂u

∂ydx+Ri,j(x, y)

∂u

∂xdy

)−∫ ba

∫ ba

(∂Ri,j(x, y)

∂x

∂u

∂x+∂Ri,j(x, y)

∂y

∂u

∂y

)dxdy = Fi,j (11)

Then by simplying Eq. (10), we can derive the finite element approximationequation gives as follows:

−∑∑

K∗i,j,r,sUr,s =∑∑

C∗i,j,r,sfr,s (12)


where,

K∗i,j,r,s =

∫ ba

∫ ba

(∂Ri,j∂x

∂Rr,s∂x

)dxdy +

∫ ba

∫ ba

(∂Ri,j∂y

∂Rr,s∂y

)dxdy,

C∗i,j,r,s =

∫ ba

∫ ba

(Ri,j(x, y)Rr,s(x, y)

)dxdy.

As a matter of fact, for the full- and half-sweep cases, the linear system inEq. (12) could be simply articulated in the form of stencil, as given below inEq. (13) and Eq. (14):

Full-sweep:

11 −4 11

Ui,j = h212

1 11 6 11 1

fi,j (13)1 1−4 01 1

Ui,j = h26

1 15 11 1

fi,j, i = 1Half-sweep:

1 10 −4 01 1

Ui,j = h26

1 11 6 11 1

fi,j, i 6= 1, n 1 10 −41 1

Ui,j = h26

1 11 51 1

fi,j, i = n (14)In fact, the stencil forms in Eqs. (13) and (14) forms consist of seven nodepoints in formulating their approximation equations. On the other hand, twoof its coefficients are zero. Apart of this, the stencil forms for both trianglefinite element schemes are the same compared to the existing five points finitedifference scheme, see [1, 21].

3 The AOR Method

The following discussion can be found in [2, 5, 12].

3.1 Formulation of 4-EGAOR Method

The following discussion can be found in [14].


3.2 Formulation of 4-EDGAOR Method

According to Abdullah [1], in solving the 2D Poisson equation via the half-sweep finite element approximation equation, the 4-EDG method proved tobe more efficient as compared to the 4-EG method. Likewise, the same stepswere adopted for the finite difference approach. Let a four solid point groupbe selected to develop a (4x4) linear system, as shown below:

4 −1 0 0−1 4 0 00 0 4 −10 0 −1 4

Ui,jUi+1,j+1Ui+1,jUi,j+1

=S1S2S3S4

(15)where,

S1 = Ui−1,j−1 + Ui−1,j+1 + Ui+1,j−1 − Fi,j+1,S2 = Ui+2,j + Ui,j+2 + Ui+2,j+2 − Fi+1,j+1,S3 = Ui,j−1 + Ui+2,j−2 + Ui+2,j+1 − Fi+1,j,S4 = Ui−1,j + Ui−1,j+2 + Ui+1,j+2 − Fi,j+1,

and,

Fi,j =h2

6(fi−2,j + fi+2,j + fi−1,j−1 + +fi−1,j+1 + fi+1,j−1 + fi+1,j+1 + 6fi,j)

The linear system in Eq. (15) can be independently decomposed into two (2x2)linear systems. Therefore, the 4-EDG method can be easily reduced as follows:[

ui,jui+1,j+1

](k+1)=

1

15

[4 11 4

] [S1S2

](16)

[ui+1,jui,j+1

](k+1)=

1

15

[4 11 4

] [S3S4

](17)

By adding the parameter, ω into Eqs. (16) and (17), the 4-EDGSOR methodcan be simplified to become:

[ui,j

ui+1,j+1

](k+1)=w

15

[4 11 4

] [S1S2

]+ (1− ω)

[ui,j

ui+1,j+1

](k)(18)

[ui+1,jui,j+1

](k+1)=w

15

[4 11 4

] [S3S4

]+ (1− ω)

[ui+1,jui,j+1

](k)(19)


where the value is defined in the range, of 1 6 ω < 2. Now, we investigate theperformance of the 4-EDGAOR method which is derived on the combinationbetween the 4-EDG and AOR methods. Therefore, by applying the AORmethod [2, 12, 21] into Eqs. (18) and (19), the general scheme for this methodcan be shown as follows:

[Ui,j

Ui+1,j+1

](k+1)=

r

15

[4T1T1

]+ω

15

[4 11 4

] [S1S2

]+ (1− ω)

[Ui,j

Ui+1,j+1

](k)(20)

[Ui+1,jUi,j+1

](k+1)=

r

15

[4T1T1

]+ω

15

[4 11 4

] [S3S4

]+ (1− ω)

[Ui+1,jUi,j+1

](k)(21)

where,

T1 = U(k+1)i−1,j−1

− U (k)i−1,j−1

+ U (k+1)i+1,j−1

− U (k)i+1,j−1

+ U (k+1)i−1,j+1

− U (k)i−1,j+1

,

S1 = U(k)i−1,j−1

+ U (k)i−1,j+1

+ U (k)i+1,j−1

− Fi,j+1,S2 = U

(k)i+2,j

+ U (k)i,j+2

+ U (k)i+2,j+2

− Fi+1,j+1,S3 = U

(k)i,j−1

+ U (k)i+2,j−2

+ U (k)i+2,j+1

− Fi+1,j,S4 = U

(k)i−1,j

+ U (k)i−1,j+2

+ U (k)i+1,j+2

− Fi,j+1.

Because of the extra benefits of the AOR method, which have two weightedparameters, all of the common existing methods become unique cases of thismethod in the scenario the parameters take certain values. For example, whenw = 1 and r = 0, we acquire the the point block Jacobi method. If w = r = 1,we acquire the point block GS method. If w = r, the point block SOR methodis attained [21].

Since, the coefficient matrix in Eq. (12) is a pentadiagonal matrix, it hasthe property A, and is Consistently Ordered [7]. At this moment, to implement4-EDGAOR method, we use Eq. (3.2) or Eq. (3.2) allows us to iterate throughhalf of the points, lying on the

√2h-grid. Again, it can be observed that

Eq. (3.2) or (3.2) involves a group of points of type •. To implement theiteration process, the algorithmn of the 4-EDGAOR method can be displayedas follows:

1. Discretize the solution domain into point of types (ie., •) as shown inFigure 1(b).


2. Perform iterations (using Eqs. (3.2) or (3.2)), taking the values of r = ωfrom the segment [1, 2).

3. Within the interval 0.1 from the value found in Step 2, define the optimalω opt with a precision of 0.01 by choosing consecutive values for whichk is minimal; r is taken to be equal to ω.

4. Perform experiments using the value of ω opt and choosing consecutivevalues of r with a precision of 0.01 within the interval 0.1 from the ω opt.

5. Define the value r opt for which k is minimal.

6. Evaluate the solutions at the remaining point of type ◦ using Eq. (13).

7. Display approximate solutions.

4 Numerical Results

In order to compare the recitals of the methods described in the previoussections, several experiments were carried out on the following 2D Poissonexample [11]:

∂2U

∂x2+

∂2U

∂y2= −(cos(x+ y) + cos(x− y)). (22)

U(x, 0) = cos x, U(x, π

2

)= 0,

U(0, y) = cos y, U(π, y) = − cos y.

and the exact solution is given by

U(x, y) = cos(x) cos(y).

The numerical experiments were carried out on a dedicated personal withan PC Intel(R) Core (TM) i7 CPU [email protected], and 6.00GB RAM. Theprogramming codes were written in C++ programming language. The AORmethod obtained in this paper is compared to other methods (4-EDGSOR,4-EDG, 4-EGAOR, 4-EGSOR and 4-EG). The value of the initial iteration isset to be zero for the test problems and in the course of implementation thetolerance error, is considered ε = 10−10. For convenience, there are three vitalparameters to be measured, including the number of iterations (k), maximumabsolute error (Abs.Error) and the execution time (t in seconds). The numer-ical results of the experiment for the proposed iterative methods are given inTable 1.


Table 1: Comparison of the number of iterations, execution time (seconds)and maximum absolute error for the iterative methods.

n Methods r w k t Abs.Error4-EG - - 60812 14.02 5.8423e-74-EDG - - 46449 73.06 4.0307e-64-EGSOR - 1.951 1766 6.91 3.3439e-7

284 4-EDGSOR - 1.959 985 3.21 4.0294e-64-EGAOR 1.969 1.972 844 4.92 2.8303e-74-EDGAOR 1.968 1.969 745 2.15 4.0293e-64-EG - - 70745 197.64 7.1419e-74-EDG - - 54099 101.64 3.4318e-64-EGSOR - 1.948 2249 10.29 2.8798e-7



356 4-EDGSOR - 1.950 2109 11.88 2.5734e-64-EGAOR 1.989 1.988 1961 16.22 2.0723e-74-EDGAOR 1.979 1.971 1105 5.57 2.5733e-6


5 Conclusions

In this paper, we have presented an application of the 4-EDGAOR methodfor solving sparse linear systems generated from the discretization of the 2DPoisson equation equations by using the Galerkin scheme. The numerical re-sults obtained for the proposed problem (Table 1) clearly show that applyingthe AOR methods reduces the number of iterations, and execution time, com-pared to the SOR and GS methods. At the same time, it has been shownthat applying the half -sweep approach reduces the computational time in theimplementation of the iterative method.

Overall, the numerical results demonstrate that the 4-EDGAOR methodoutperforms the existing block methods (4-EDGSOR, 4-EDG, 4-EGAOR, 4-EGSOR and 4-EG), particularly in the sense of the number of iterations andexecution time. This is mainly attributable to the reduction of the computa-tional complexity; since the implementations of the 4-EDGAOR method onlyconsider approximately half of all interior node points in a solution domain. Forfuture work, the capability of the quarter-sweep approach [2, 12, 21] should beinvestigated in terms of the point block iterative method by using the Galerkinscheme.

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Received: October 4, 2015; Published: December 2, 2015

numerical performance of triangle element approximation ... · 4-edgaor method was implemented by...

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