A Computational Study with Finite Element Method and Finite Difference Method for 2D Elliptic Partial Differential Equations

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1 Applied Mathematics, 05, 6, 04-4 Pblished Online November 05 in SciRes. A Comptational Stdy with Finite Element Method and Finite Difference Method for D Elliptic Partial Differential Eqations George Papanikos, Maria Ch. Gosido-Kotita School of Mathematics, Aristotle University, Thessaloniki, Greece Received September 05; accepted 7 November 05; pblished 0 November 05 Copyright 05 by athors and Scientific Research Pblishing Inc. This work is licensed nder the Creative Commons Attribtion International License (CC BY). Abstract In this paper, we consider two methods, the Second order Central Difference Method (SCDM) and the Finite Element Method (FEM) with P trianglar elements, for solving two dimensional general linear Elliptic Partial Differential Eqations (PDE) with mied derivatives along with Dirichlet and Nemann bondary conditions. These two methods have almost the same accracy from theoretical aspect with reglar bondaries, bt generally Finite Element Method prodces better approimations when the bondaries are irreglar. In order to investigate which method prodces better reslts from nmerical aspect, we apply these methods into specific eamples with reglar bondaries with constant step-size for both of them. The reslts which obtained confirm, in most of the cases, the theoretical reslts. Keywords Finite Element Method, Finite Difference Method, Gass Nmerical Qadratre, Dirichlet Bondary Conditions, Nemann Bondary Conditions. Introdction Finite Difference schemes and Finite Element Methods are widely sed for solving partial differential eqations []. Finite Element Methods are time consming compared to finite difference schemes and are sed mostly in problems where the bondaries are irreglar. In particlarly, it is difficlt to approimate derivatives with finite difference methods when the bondaries are irreglar. Moreover, Finite Element methods are more complicated than Finite Difference schemes becase they se varios nmerical methods sch as interpolation, nmerical integration and nmerical methods for solving large linear systems (see []-[6]). Also the mathematically deriva- How to cite this paper: Papanikos, G. and Gosido-Kotita, M.Ch. (05) A Comptational Stdy with Finite Element Method and Finite Difference Method for D Elliptic Partial Differential Eqations. Applied Mathematics, 6,

2 tion of Finite Element Methods come from the theory of Hilbert Space, and Sobolev Spaces as well as from variational principles and the weighted residal method (see [5]-[]). In this paper, we will describe the Second order Central Difference Scheme and the Finite Element Method for solving general second order elliptic partial differential eqations with reglar bondary conditions on a rectanglar domain. In addition, for both of these methods, we consider the Dirichlet and Nemann Bondary conditions, along the for sides of the rectanglar area. Also, we make a brief error analysis for Finite element method. Moreover, for the finite element method, we site two other important nmerical methods which are important in order that the algorithm can be performed. These methods are the bilinear interpolation over a linear Lagrange element, Gass qadratre and contor Gass Qadratre on a trianglar area. Frthermore, these two schemes lead to a linear system which we have to solve. For the prpose of this paper, we solve the otcome systems with Gass-Seidel method which is briefly discssed. In the last section, we contacted a nmerical stdy with Matlab R05a. We apply these methods into specific elliptical problems, in order to test which of these methods prodce better approimations when the Dirichlet and Nemann bondary conditions are imposed. Or reslts show s that the accracy of these two methods depends on the kind of the elliptical problem and the type of bondary conditions. In Section, we stdy the Second order Central Difference Scheme. In Section, we give the Finite Element Method, bilinear interpolation in P, Gass Qadratre, Finite Element algorithm and error analysis. In Section 4, we give some nmerical reslts, in Section 5, we give the conclsions and finally in Section 6 we give the relevant references.. Second Order Central Difference Scheme The second order general linear elliptic PDE of two variables and y given as follow: p + s + q + b + b + r = f y () Ω= ab, cd,, it holds with defined on a rectanglar domain [ ] [ ] pqsb,,,, b, r, f C ( Ω ) and C( Ω) C ( Ω) Moreover in this paper two types of bondary conditions are considered: y (, ) = g( y, ) on Γ (Dirichlet Bondary Conditions). = g( ) on Γ (Nemann Bondary Conditions). n The bondary Ω = Γ Γ and n is the normal vector along the bondaries. We divide the rectanglar domain Ω in a niform Cartesian grid ( i j) ( ) s 4pq < 0. Also, y = i h, j k : i =,,, N, j =,,, M where N, M are the nmbers of grid points in and y directions and b d h = and k N = M are the corresponding step sizes along the aes and y. The discretize domain are shown in Figre. Using now the central difference approimation we can approimate the partial derivatives of the relation () as follows: + h i+, j i, j i, j = + O h + = + O k + y 4hk ( h ) i+, j+ i, j+ i+, j i, j () () + i, j+ i, j i, j = + O k k (4) 05

3 Figre. Discrete domain. where O( h ), O( k ) and O( k h ) = + O h h i+, j i, j = + O k k i, j+ i, j + are the trncation errors. We now approimate the PDE () sing the relations (), (), (4), (5), (6) and we obtain the second order central difference scheme: 4 hkr k p hq + k p + b h + k p b h i, j i, j i, j i, j i, j i, j i+, j i, j i, j i, j + h q + b k + h q b k + s hk + = 4h k f i, j i, j i, j+ i, j i, j i, j i, j i+, j+ i, j+ i+, j i, j i, j With trncation error O( k h ) +. The relation (7) can be written as a linear system: (5) (6) (7) A = b (8) Dirichlet Bondary Conditions The dimensions of the above linear system depends on the bondary conditions. More specific, if we have the Dirichlet Bondary Conditions: = gay, = gay, for each j= 0,,, M 0, j j N, j j = g, c = g, d for each i = 0,,, N i,0 i im, i 06

4 then the dimensions of the matri A, and b are: ( N )( M ) ( N )( M ) ( N )( M ) for the vectors, b and for the matri A. Moreover, the form of matri A and the vector are given by: and B D O O O O O O G B D O O O O O O G B D O O O O A = O O O O CM BM DM O O O O O CM BM DM O O O O O GM BM =,,,,,,, N,,,,,,,,, N,,,, M,, N, M As we can see the matri A is tri-diagonal block Matri. These block matrices are tri-diagonal as well of order ( N )( N ) Nemann Bondary Conditions B, k =,,, M ; G, l =,,, M ; D, m=,,, M k l m ( d, ) = g( ) ( c, ) = g ( ) ( b, y) = g ( y) ( a, y) = g ( y) 4 G. Papanikos, M. Ch. Gosido-Kotita We approimate the Nemann bondary conditions in every side of the rectanglar domain as follows st side (North side of the rectanglar area) im,, (, d ) g ( + im = ) = g i im, + = im, + kg i for j = M, i =,,, N (9) k nd side (East side of the rectanglar area) N+, j N, j ( b, y ) = g ( y ) = g j N+, j = N, j + hg j for j =,,, M, i = N (0) h rd side (Soth side of the rectanglar area) i,, (, c ) g ( i = ) = g i i, = i, kg i for j = 0, i =,,, N () k 4 th side (West side of the rectanglar area), j, j ( a, y ) = g 4 ( y ) = g 4 j, j =, j hg 4 j for j =,,, M, i = 0 () h Using the relations (9), (0), (), () the vales im, +, N+, j, i, and, j which lies otside the rectanglar domain can be eliminated when appeared in the linear system. Ths the block tri-diagonal matri A has dimensions ( N + )( M + ) ( N + )( M + ) and the vectors, b are N + M + order. The matri A and the vector are given below: of 07

5 and B0 L0 O O O O O O O C K D O O O O O O O C K D O O O O O O O C K D O O O O A = O O O O O CM KM DM O O O O O O O CM KM DM O O O O O O O LM BM = 0,0,,0,,0,, N,0, 0,,,,,,, N,,, 0, M,, N, M, NM, where Bl, Ll, l = 0, M and Ck, Kk, Dk, k =,,, M M +. In order to solve the linear system (8), we se the Gass-Seidel method (GSM) (see [] []). An important property that the matri A mst have is to be strictly diagonally dominant in order the GCM to converge. Theorem ( 0) ( k ) If A is strictly diagonally dominant, then for any choice of, Gass-Seidel method give seqence { } k = 0 that converge to the niqe soltion of A = b. The proof of theorem can be fond in [] [].. Finite Element Method In this section we consider an alternative form of the general linear PDE () are tri-diagonal matrices with dimensions s s p q + c + d + r = f where p C ( Ω), q C ( Ω), s C ( Ω), c C ( Ω), d C ( Ω), r C( Ω), f C( Ω ) and C ( Ω) C( Ω) (). With bondary conditions y (, ) = g( y, ) on Γ (Dirichlet Bondary Conditions). = g( ) on Γ (Nemann Bondary Conditions). n And the bondary Ω = Γ Γ In order to approimate the soltion of () with FEM algorithm we mst transform the PDE into its weak form and solve the following problem. Ω H Γ Find where and (, ) av = l v v H Γ Ω (4) v v s v s v a(, v) = p + q + + c vd vrv dd y Ω are bilinear and linear fnctionals as well. lv = fvy dd + gvs d Ω It is sfficient now to consider that L ( Ω ),, L Γ Ω. Also we assme that 08

6 ( Ω) ( Ω ) and g L ( Γ ) pqscdr,,,,, L, f L when the Nemann bondary Conditions are applied Γ gvds 0, else if we have only Dirichlet Bondary conditions then the line integral is eqal to zero. The finite element method approimates the soltion of the partial differential Eqation () by minimizing the fnctional: v v v v v v J v p q s c v d v rv fv y gvs v H Ω Γ = dd d Γ ( Ω) (5) { } where HΓ ( Ω ) = Η ( Ω ) = g on Γ and { L : D L} the weak derivatives of. The spaces Η ( Ω ), Η Ω = Ω Ω. Also with D we denote H Γ are Sobolev fnction spaces which also considered to be Hilbert spaces(see [7]-[9]). The niqeness of the soltion of weak form (4) depends on La-Milgram theorem along with trace theorem (see [7]). In addition according to Rayleigh-Ritz theorem the soltion of the problem (4) are redced to minimization of the linear fnctional : J H Γ Ω, (see [7]). The first step in order the FEM algorithm to be performed is the discretization of the rectanglar domain Ω= [ ab, ] [ cd, ] R by sing Lagrange linear trianglar elements. We denote with P k the set of all polynomials of degree k in two variables [7]. For k = we have the linear Lagrange triangle and { ϕ C ϕ( y) a b cy} = Ω,, = + +, dim = Also the trianglation of the rectanglar area shold have the below properties: We assme that the trianglar elements Ti, i κκ, = κ( h), are open and disjoint, where h is the maimm diameter of the triangle element. The vertices of the triangles all call nodes, we se the letter V for vertices and E for nodes. We also assme that there are no nodes in the interior sides of triangles... Bilinear Interpolation in P Let as consider now the trianglation of the rectanglar domain Ω= [ ab, ] [ cd, ] R as we describe to a previos section. In every triangle T i of the domain we interpolate the fnction with the below linear polynomial: with interpolation conditions: ( i ) (, ) ϕ y = a + b + cy ( i ) j ( j j) ( j j) ϕ, y =, y, j =,, i i in every verte Vj = j, yj of a trianglar element. Ths it creates the below linear system with nknown coefficients a, b, c. i ( i) ( i), i ( i) ( i), = i ( i) ( i), y y ϕ y y a ϕ y y b ϕ c Solving the system we find the approimate polynomial of where ( i ) ( i ) i i i i i i i y, = N y,, y + N y,, y + N y,, y ϕ ϕ ϕ ϕ = N y, ϕ, y j= j j j j 09

7 and i ( i) ( i) ( i) i ( i) ( i) ( i) = + + i ( i) ( i) ( i) = + + (, ) (, ) (, ) N y = a + b + c y N y a b c y N y a b c y ( i) ( i) ( i) ( i) ( i) ( i) ( i) ( i) y y y y i i i =, =, = a b c A A A ( i) ( i) ( i) ( i) ( i) ( i) ( i) ( i) y y y y i i i =, =, = a b c A A A ( i) ( i) ( i) ( i) ( i) ( i) ( i) ( i) y y y y i i i =, =, = a b c A A A i i i i The fnction N j y, = a + b + c yis the interpolation fnction or shape fnction and it has the following property:.. Gass Qadratre ( i ) Α ν j = k N j ( k, yk) =, k =,, 0 Αν j k An important step in order to implement the Finite Element algorithm is to compte nmerically the doble and line integrals which occrs in every trianglar element (see [5] []). Gass Qadratre in Canonical Triangle As a canonical triangle we consider the triangle with vertices (0, 0), (0, ) and (, 0) and we denote T= y, :0, + y. The approimation rle of the doble integral in canonical triangle is given below: κ { } Tκ n g f y y w f y f y P n y (, ) dd i ( i, i), (, ) κ g ( i, i) Where n g is the nmber of Gass integration points, w i are the weights and (, ) i= i y i are the Gass integration points. The linear space P κ is the space of all linear polynomial of two variables of order k The following Table gives the nmber of qadratre points for degrees to 4 as given in Ref. [0]. It shold be mentioned that for some N, the corresponding ng is not necessarily niqe. (see [0] and references therein). Gass qadratre in general trianglar element Initially we transform the general triangle Τ into a canonical triangle sing the linear basis fnctions: Table. Qadratre points for degrees to 4. ( ξη, ) ( ξη, ) Ν ξη, = ξη Ν = ξ Ν = η Qadratre points for degrees to 4 N dim( N ) n g

8 with The variables, y for the random triangle can be written as affine map of basis fnctions: ( ξη, ) i( ξη, ) i ( ξη, ) ( ξη, ) ( ξη, ) = r = Ν =Ν +Ν +Ν i= ( ξη, ) i( ξη, ) i ( ξη, ) ( ξη, ) ( ξη, ) y = r = Ν y =Ν y +Ν y +Ν y i= Also we have the Jacobian determinant of the transformation J ( y, ) ξ ξ, = = = Ak ( ξη, ) η η ( ξη) Using the above relations we obtain the Gass qadratre rle for the general trianglar element: T n g (, ) dd k i ( ( ξi, ηi), ( ξi, ηi) ) I = F y y A wf r r A k = i= ( ) + ( ) + ( ) y y y y y y is the area of the triangle. Contor qadratre rle In the Finite Element Method when the Nemann bondary conditions are imposed it is essential to compte nmerically the below Contor integral in general trianglar area. Pj I= g y, d s= g y, ds The basic idea is to transform the straight contor P i P j to an interval l [ ab, ] =, and then the Gassian qadratre for single variable fnction. Using the basis fnctions we have the following relations in every side of the triangle Along side (P P ): P P Along side : (P P ): P P = ( ) + ( ) ( + ( ) + ( ) ) g, y d s y y g ξ, y y y ξ dξ 0 Pi ( ) + y y ( )( + ξ) ( y y)( + ξ) g y = +, + dξ ( ) ( y y ) N ( )( + ξi) ( y y)( + ξi) cg i, y i= = ( ) + ( ) ( + ( ) + ( ) ) g, y d s y y g η, y y y η dη 0 ( ) y y ( )( + η) ( y y)( + η) g y + =, dη + + ( ) ( y y ) N ( )( + ηi) ( y y)( + ηi) cg i, y i=

9 Along side : (P P ): P P = ( ) + ( ) ( + ( ) + ( ) ) g, y d s y y g η, y y y η dη 0 ( ) y y ( )( + η) ( y y)( + η) g y + =, dη + + ( ) ( y y ) N ( )( + ηi) ( y y)( + ηi) cg i, y i= The error of the bilinear interpolation Gass qadratre depend on the dimension of the polynomial sbspace (see [])... Finite Element Algorithm The Finite element algorithm has the prpose to find the approimate soltion of the problem (5) in a sbspace of H Γ. We consider as sbspace the of all piecewise linear polynomials with two variables polynomials of degree one. i.e.: ( i ) (, ) ϕ y = a + b + cy The inde i represents the nmber of trianglar elements which eist in the rectanglar domain. The polynomials mst be piecewise becase the linear combination of them mst form a continos and integrable fnction with continos first and second derivatives. The eistence and niqeness of the approimate soltion is ensred by the La-Milgram-Galerkin and Rayleight- Ritz theorems (see [] [7] [8]). Firstly as we describe in a previos section, we have to trianglate the domain before the algorithm evalated. After that the algorithm seeks approimation of the soltion of the form: m (, ) γϕ (, ) y = y h i i i= where ϕ i, i =,,, m is the linear combination of independent piecewise linear polynomials and γ i, i =,,, m are constants with m is the nmber of nodes. Actally, the polynomials ϕ i corresponds to shape fnctions Ν j, j =,, in every verte of the triangles. Ths the approimate soltion is the linear combination of all the independent interpolation fnctions mltiplied with some constant γ i. Some of these constants for eample, γn+, γn+,, γm are sed to ensre that the Dirichlet bondary conditions if there are eist y, = g y, h are sed to minimize the fnctional ( h ) m Inserting the approimate soltion ( y, ) = γϕ ( y, ) for v into the fnctional are satisfied on Γ and the remaining constants γ, γ,, γn h i i i= J. J v and we have: m m m m m m m ϕi ϕi ϕi ϕi ϕi J γϕ i i = p γi + q γi + s γi γi c γi γϕ i i i= Ω i= i= i= i= i= i= m m m ϕ m m i d γi γϕ i i r γϕ i i + f γϕ i i dd y d i= g γϕ i i s i= i= i= Γ i= Consider J as a fnction of γ, γ,, γn. For minimm to occr we mst have J γ j = 0, j =,,, n (6)

10 Differentiating (6) gives n ϕ ϕ i j s ϕ ϕ i j s ϕ ϕ i j ϕ ϕ i j p q? i= Ω c ϕ ϕ i j d ϕ ϕ i j ϕj + ϕi ϕj + ϕi rϕϕ i j dd yγi ϕ ϕ s ϕ ϕ s ϕ ϕk = m i k i k i fϕjdd y gϕjds p Ω Γ k= n+ Ω y ϕi ϕk c ϕi ϕk d ϕi ϕk + q ϕk + ϕi ϕk + ϕi rϕϕ i k dd yγk for each j =,,, n.this set of eqations can be written as a linear system: t where = (,,, ) c γ γ γ n, A ( a ij ) for each i =,,, n, j =,,, m. t = and = (,,, ) A c = b (7) b β β β n are defined by ϕ ϕ i j s ϕ ϕ i j s ϕ ϕ i j ϕ ϕ i j aij = p q Ω c ϕ ϕ i j d ϕ ϕ i j ϕj + ϕi ϕj + ϕi rϕϕ i j dd y m ϕi ϕk s ϕi ϕk s ϕi ϕk β = fϕ dd y+ gϕ ds p + + y y i j j Ω Γ k= n+ Ω ϕi ϕk c ϕi ϕk d ϕi ϕk + q ϕk + ϕi ϕk + ϕi rϕϕ i k dd yγk The choice of sbspace for or approimation is important becase can ensre s that the matri A will be positive definite and band( []-[4]). According to the previos analysis leads again to a linear system, which can be solved as we described in a previos section with Gass-Seidel Method (see []-[4])..4. Error Analysis Let s consider again the problem (4) Ω : Find H Γ (, ) The approimation of finite element of the problem (8) is given below: Find h : av = l v v H Γ Ω (8) (, ) h h h h h Cea s lemma The finite element approimation h i.e: H Γ ( Ω) c min v a v = l v v (9) of the weak soltion H Γ h H ( Ω) h H Γ c vh V Ω h Γ 0 Ω is the best fit to in the norm The error analysis of finite element method depend on the Cea s lemma for elliptic bondary vale problems. The proof can be fond in [] [7].

11 Now we will present withot proof the following statement [7]. s min v h C h (0) v V h h H Γ Ω where C( ) is positive constant dependent on the smoothness of the fnction, h is the mesh size parameter and s is a positive real nmber, dependent on the smoothness of and the degree of the piecewise polynomials comprising in. In or case we have the Lagrange linear elements so the degree of the piecewise polynomials is one. Combining, relations Cea s lemma we shall be able to dedce that: c () h C h H Γ c0 Ω The relation () gives a bond of the global error e= h in terms of the size mesh parameter h. Sch a bond on the global error is called priori error bond. L -norm For proving an error estimate in L -norm the reglarity of the soltion of () plays an essential role. By the Abin-Nitsche dality argment the error estimate in L norm between and its finite element approimation h O h, (see [] [7]). is O( h ). However this bond can be improved to 4. Nmerical Stdy In this section we contact a nmerical stdy sing Matlab R05a. For the prpose of this paper we cite representative eamples of second order general elliptic partial differential eqations in order to make comparisons between these two methods with varios step-sizes and the mesh size parameters of finite element method. Ths in each eample we present reslts for the absolte and relevant absolte errors in L norm along with their graphs. Also we make graphical representations of the eact and approimate soltion of the specific problem as well. The problems of the eamples can be fond in [] [4]. Eample Find the approimate soltion of the partial differential eqation + = 0, 0 4, 0 y 4 with Dirichlet bondary conditions along the rectanglar domain and eact soltion (, ) e y cos e cos, (, ) y = y y Ω, e y y = cos e cosy Reslts (Table and Table ) In Figre and Figre we have the graphs for SCDM and in Figre 4 and Figre 5 for FEM. Eample Find the approimate soltion of the partial differential eqation + = f( y, ),0,0 y 0 with Dirichlet bondary conditions along the rectanglar domain = 0 on three lower side of Ω and Nemann bondary condition (, ) = 0 and eact soltion πy y (, ) = sin ( π) sin 4

12 Table. Nmerical reslts. Second order central difference scheme with steps h= 0., k = 0. Finite element method with mesh size h = 0. y Absolte error L L 00% L Absolte error L L 00% L e6.74e e Table. Nmerical reslts. Second order central difference scheme with steps h= 0.05, k = 0.05 Finite element method with mesh size h = 0.05 y Absolte error L L 00% L Absolte error L L 00% L e5.567e e e e e

13 Figre. SCDM graphs. Figre. SCDM graphs. 6

14 Figre 4. FEM graphs. Approimate soltion with h = 0.05 Eact soltion with h = 0.05 Figre 5. FEM graphs. 7

15 Reslts (Table 4 and Table 5) In Figre 6 and Figre 7 we have the graphs for SCDM and in Figre 8 and Figre 9 for FEM. Eample Find the approimate soltion of the partial differential eqation + ( + y) ( + y+ y) = f( y, ), 0, 0 y with Dirichlet bondary conditions along the rectanglar domain and eact soltion Table 4. Nmerical reslts. y 0, y = 0.50e, y = 0.50e y+ + (, 0) = 0.50e (,) = 0.50 e + log ( ) (, ) 0.50 e + y = + log ( + )( ) y y Second order central difference scheme with steps h= 0., k = 0. Finite element method with mesh size h = 0. y Absolte error L L 00% L Absolte error L L 00% L Table 5. Nmerical reslts. Second order central difference scheme with steps h= 0.05, k = 0.05 Finite element method with mesh size h = 0.05 y Absolte error L L 00% L Absolte error L L 00% L e

16 Figre 6. SCDM graphs. Figre 7. SCDM graphs. 9

17 Figre 8. FEM graphs. Approimate soltion with h = 0.05 Eact soltion with h = 0.05 Figre 9. FEM graphs. 0

18 Reslts (Table 6 and Table 7) In Figre 0 and Figre we have the graphs for SCDM and in Figre and Figre for FEM. Overall, what stands ot from these eamples is that the finite element method has better approimations for the first problem compared to finite difference method for all the step-sizes that we have. Bt in the second problem we have a small difference in the reslts with better accracy for the finite difference method and in the third problem the finite element method has bigger relevant errors than the difference method. More specifically, in eample according to the tables we have better approimations for the finite element method in both of step sizes and the mesh size parameters to specific points bt the graphs of the percentage of relevant errors sggest that the second order central finite difference scheme prodce better approimations generally. On the other hand, in the other two eamples according to the tables and the graphs of errors in the second problem we have a small difference between these methods and in the third problem we have better approimations of the second order central difference scheme almost in all points of the domain. Conclsively, we can notice that in the third eample for both of these methods the reslts which we obtained are almost identical when different step sizes are applied in CFDM and mesh size parameters in FEM. Table 6. Nmerical reslts. Second order central difference scheme with steps h= 0., k = 0. Finite element method with mesh size h = 0. y Absolte error L L 00% L Absolte error L L 00% L e5 4.69e Table 7. Nmerical reslts. Second order central difference scheme with steps h= 0.05, k = 0.05 Finite element method with mesh size h = 0.05 y Absolte error L L 00% L Absolte error L L 00% L e

19 Figre 0. SCDM graphs. Figre. SCDM graphs.

20 Figre. FEM graphs. Figre. FEM graphs.

21 5. Conclsion Finally, we can say that the data which we obtained from these eamples show that both of these methods prodce qite sfficient approimations for or problems. Also the reslts prove that the accracy of them depends on the kind of the elliptical problem and the type of bondary conditions. For frther research, the approimations of these methods can be improved. This improvement can be made if in the second order difference scheme we keep more taylor series terms in order to approimate the derivatives and in finite element method if we se higher order elements sch as qadratic Lagrange trianglar elements or cbic Hermite trianglar elements. References [] McDonogh, J.M. (008) Lectres on Comptational Nmerical Analysis of Partial Differential Eqations. [] Gosido-Kotita, M.C. (04) Comptational Mathematics. Tziolas, Ed. [] Gosido-Kotita, M.C. (009) Nmerical Methods with Applications to Ordinary and Partial Differential Eqations. Notes, Department of Mathematics, Aristotle University of Thessaloniki, Thessaloniki. [4] Gosido-Kotita, M.C. (004) Nmerical Analysis. Christodolidis, K., Ed. [5] Dnavant, D.A. (985) High Degree Efficient Symmetrical Gassian Qadratre Rles for the Triangle. International Jornal for Nmerical Methods in Engineering,, [6] Reynolds, A.C., Brden, R.L. and Faires, J.D. (98) Nmerical Analysis. Second Edition. Yongstown State University, Yongstown. [7] Dogalis, V.A. (0) Finite Element Method for the Nmerical Soltion of Partial Differential Eqations. Revised Edition, Department of Mathematics, University of Athens, Zografo and Institte of Applied Mathematics and Comptational Mathematics, FORTH, Heraklion. [8] Sli, E. (0) Lectre Notes on Finite Element Method for Partial Differential Eqations. Mathematical Institte, University of Oford, Oford. [9] Everstine, G.C. (00) Nmerical Soltion of Partial Differential Eqations. George Washington University, Washington DC. [0] Szabó, B. and Babska, I. (99) Finite Element Analysis. Wiley, New York. [] Isaacson, E. and Keller, H.B. (996) Analysis of Nmerical Methods. John Wiley and Sons, New York. [] Papanikos, G. (05) Nmerical Stdy with Methods of Characteristic Crves and Finite Element for Solving Partial Differential Eqations with Matlab. Master s Thesis. [] Yang, W.Y., Cao, W., Chng, T.S. and Morri, J. (005) Applied Nmerical Methods Using MATLAB. John Wiley & Sons, Hoboken. [4] Gropp, W.D. and Keyest, D.E. (989) Domain Decomposition with Local Refinement. Research Report YALEU/DCS/ RR-76. 4