An Heuristic Finite Difference Scheme on Irregular Plane Regions

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1 Applied Mathematical Sciences, Vol 8, 2014, no 14, HIKARI Ltd, wwwm-hikaricom An Heuristic Finite Difference Scheme on Irregular Plane Regions Francisco J Domínguez-Mota, Pablo M Fernández-Valdez Erika Ruiz-Díaz, Gerardo Tinoco-Guerrero and José G Tinoco-Ruiz Universidad Michoacana de San Nicolás de Hidalgo Facultad de Ciencias Físico Matemáticas Edificio B, Ciudad Universitaria Morelia, México Copyright c 2014 Francisco J Domínguez-Mota et al This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited Abstract In this paper, we present an heuristic finite difference scheme for the second-order linear operator, which is derived from an unconstrained least squares problem defined by the consistency condition on the residuals of order one, two and three in the Taylor expansion of the local truncation error It is based on a non-iterative calculation of the difference coefficients and can be used to solve efficiently Poisson-like equations on non-rectangular domains which are approximated by structured convex grids Mathematics Subject Classification: 65M06, 65M50 Keywords: finite differences, Poisson-like equations, irregular regions, convex structured grids 1 Introduction A very important problem in scientific computing is the numerical solution of Poisson-like problems It is a well studied problem when the domain has a simple geometry; unfortunately, in some specific real-world settings, the

2 672 F Domínguez-Mota et al Figure 1: Example of a typical domain when m = n =4 domain is irregular and very non-symmetric, and in this case the classical finite difference formulas do not produce accurate results As a matter of fact, there are rather few reliable finite difference schemes which can be successfully applied on irregular domains In the following sections we present a sound heuristic scheme based on a least squares problem for this case The domains of interest here are simply connected polygonal domains -mostly irregular- which can not be decomposed into rectangles For such domains, it is possible to generate suitable convex structured grids using the direct optimization method, as discussed in detail in [3, 10, 12, 13, 16] To introduce the required notation for the grids, let m and n be the number of vertical and horizontal numbers of nodes on the sides of a typical domain boundary; the latter is the positively oriented polygonal Jordan curve γ of vertices V = {v 1,,v 2m+n 2) }, and it defines the typical domain Ω IR 2 An example can be seen in figure 1 A set G = {P i,j 1 i m, 1 j n} of points of the plane with the fixed boundary positions given by V is a structured grid with quadrilateral elements for Ω, of order m n; a grid G is convex if and only if each one of the m 1)n 1) quadrilaterals or cells) c i,j of vertices {P i,j,p i+1,j,p i,j+1,p i+1,j+1 },1 i<m,1 j<n, is convex and non-degenerate Fig 2) The functional which was minimized to generate the convex structured grids of the numerical tests for this paper, as implemented in UNAMALLA [4], was

3 An heuristic finite difference scheme 673 G Figure 2: Example of a typical grid G with 4 points per side for the boundary shown in figure 1 the convex linear combination of the area functional S ω G) and length LG) with weight σ =05 See [11] for details) 2 Finite difference schemes for linear operators Let us consider the second order linear operator Lu = Au xx + Bu xy + Cu yy + Du x + Eu y + Fu 1) where A, B, C, D, E and F are smooth functions defined on an open set containing Ω; general finite difference schemes can be obtained by considering a finite set of nodes p 0 =x 0,y 0 ),, p k =x k,y k ), for which it is required to find coefficients Γ 0, Γ 1, Γ 2,, Γ k such that Γ i up i ) [Au xx + Bu xy + Cu yy + Du x + Eu y + Fu] p0 2) i=0 It is important to emphasize that there are few efficient finite difference schemes for numerical solution of partial differential equations on non-rectangular regions using finite differences [1, 2, 5, 6, 7, 9, 14] The limit value in 2) is given by the weak consistency condition [15] Γ i up i ) [Au xx + Bu xy + Cu yy + Du x + Eu y + Fu] p0 0, as p 1, p q p 0, i=0

4 674 F Domínguez-Mota et al whose Taylor expansion up to third order yields [Au xx + Bu xy + Cu yy + Du x + Eu y + Fu] p0 Γ i up i ) = i=0 ) ) F p 0 ) Γ i up 0 )+ Dp 0 ) Γ i Δx i u x p 0 ) + i=0 ) Γ i Δx i ) 2 ) Ep 0 ) Γ i Δy i u y p 0 )+ Ap 0 ) u xx p 0 ) + 2 ) Γ i Δy i ) 2 ) Bp 0 ) Γ i Δx i Δy i u xy p 0 )+ Cp 0 ) u yy p 0 ) + 2 Γ i Δx i ) 3 ) Γ i Δx i ) 2 ) Δy i u xxx p 0 )+ u xxy p 0 ) + 3! 2 Γ i Δx i Δy i ) 2 ) Γ i Δy i ) 3 ) u xyy p 0 )+ Γ i u yyy p 0 ) + 2 3! O max{δx i, Δy i }) 4 where Δx i = x i x 0, Δy i = y i y 0,1 i k In the case of the 3 3 grid defined by p 0,p 1,, p 8, the consistency condition up to third order defines the linear system Δx 1 Δx 8 0 Δy 1 Δy 8 0 Δx 1 ) 2 Δx 8 ) 2 0 Δx 1 Δy 1 Δx 8 Δy 8 0 Δy 1 ) 2 Δy 8 ) 2 0 Δx 1 ) 3 Δx 8 ) 3 0 Δx 1 ) 2 Δy 1 Δx 8 ) 2 Δy 8 0 Δy 1 ) 2 Δx 1 Δy 8 ) 2 Δx 8 0 Δy 1 ) 3 Δy 8 ) 3 Γ 0 Γ 1 Γ 2 Γ 8 = F p 0 ) Dp 0 ) Ep 0 ) 2Ap 0 ) Bp 0 ) 2Cp 0 ) which must be satisfied by the Γ i coefficients One must note that, in general, this system is not well-determined Hence, the main question is how to produce a solution of 3) which provides a consistent scheme In [8], in order to get a second order scheme, we proposed to calculate the coefficients Γ 0, Γ 1,, Γ 8 by solving the constrained optimization problem 3) min z = R R2 7 + R2 8 + R2 9, subject to R i =0,i=0,, 5 4)

5 An heuristic finite difference scheme 675 where R 0 = 8 i=0 Γ i F, R 3 = 8 Γ i Δx i ) 2 2A, R 1 = 8 Γ i Δx i ) D, R 4 = 8 Γ i Δx i )Δy i ) B, R 2 = 8 Γ i Δy i ) E, R 5 = 8 Γ i Δy i ) 2 2C, 5) R 6 = 8 Γ i Δx i ) 3 R 7 = 8 Γ i Δx i ) 2 Δy i ) R 8 = 8 Γ i Δx i )Δy i ) 2 R 9 = 8 Γ i Δy i ) 3 Problem 4) must be solved iteratively for the 3 3 structured subgrid around every inner grid node P i,j defined by G to provide the optimal local approximation to the differential operator 1) Very satisfactory results obtained with this scheme were reported in [8] In this paper, we propose an alternative heuristic scheme based on an unconstrained optimization problem which is closely related to the constrained one First, we separate the first equation of the matrix system 3) 8 Γ i F = 0 7) and then we solve the least squares problem defined by Γ Δx 1 Δx 1 8 Dp Γ 0 ) Δy 1 Δy 2 8 Ep Δx 1 ) 2 Δx 8 ) 2 0 ) = 2Ap 0 ) 8) Δx 1 Δy 1 Δx 8 Δy 8 Bp Δy 1 ) 2 Δy 8 ) 2 0 ) 2Cp Γ 0 ) 8 through the Cholesky factorization of its normal equations 6) M T M Γ=M T β, 9) where M = Δx 1 Δx 8 Δy 1 Δy 8 Δx 1 ) 2 Δx 8 ) 2 Δx 1 Δy 1 Δx 8 Δy 8 Δy 1 ) 2 Δy 8 ) 2 Γ= Γ 1 Γ 2 Γ 8, β = Dp 0 ) Ep 0 ) 2Ap 0 ) Bp 0 ) 2Cp 0 ) Next, Γ 0 is obtained from 7) One must note that, due to the lack of restrictions, this scheme is indeed less

6 676 F Domínguez-Mota et al restrictive than the previous one, and that in some very irregular grids, the residuals of order one and two might not be equal to zero; however, in spite of these facts, a large amount of numerical experiments led us to conclude that the proposed scheme is accurate enough In a similar manner to the rectangular case, the approximation to the differential operator at every inner grid point produces a linear equation whose coefficients are given by the Γ i values, and the set of all these equations can be rearranged into a sparse matrix-vector algebraic system of equations whose solution approximates the solution of 1) 3 Numerical Test For the numerical tests, we have selected 4 polygonal domains which resemble geographical regions: Great Britain ENG), Havana bay HAB), Michoacán MIC) and Ucha UCH) Using these boundaries, we considered two sets of tests: a) Convex grids with 21 points per side were generated in UNAMALLA by minimizing the functional [4] 1/2S ω G)+LG)), these grids were lated uniformly refined in order to generate grids with 41 and 81 points per side The refined grids with 41 points per side for these regions are shown in figure 3 b) Convex grids with 21, 41 and 81 points per side were generated in UNA- MALLA by minimizing the same functional The optimal grids with 41 points per side for these regions are shown in figure 4 The resulting structured grids were used to calculate the Γ i coefficients for the schemes described in the previous section In all cases, the algebraic systems obtained from 1) were assembled and later solved using a sparse Gaussian Elimination routine in MATLAB 72 in a personal computer with Intel Pentium processor B970@230 GHz and 6Gb of RAM memory To test the performance of the new scheme in the solution to the Poisson problem K u) =f, the following values for K and u were selected [5]: 1 First problem Kx, y) = ), u = 2 exp2x + y)

7 An heuristic finite difference scheme 677 Michoacán Havana Bay Ucha Great Britain Figure 3: Refined grids with 41 points per side

8 678 F Domínguez-Mota et al Michoacán Havana Bay Ucha Great Britain Figure 4: Optimal grids with 41 points per side

9 An heuristic finite difference scheme Second problem with and D = P = Kx, y) =P T DP, cos π 8 ) sin π 8 ) sin π 8 ) cos π 8 ) 1+2x 2 + y x 2 +2y 2 u = sinπx) sinπy) Function f inside the domain was chosen in such a way that u was the exact solution in both cases Moreover, for the boundary, the function u was selected as the Dirichlet condition The tests results are summarized in tables 1 and 2 for the tests set a, and in tables 3 and 4 for the test set b In all the tables, the values of the quadratic error norm is calculated as the grid function u U 2 = u i,j U i,j ) 2 A i,j, i,j where U and u are the approximated and the exact solution calculated at the i, j th grid point P i,j =x i,j,y i,j ) respectively, and A i,j is the area of the i, j th - element, calculated as the area of the polygon defined by {P i+1,j,p i,j+1,p i 1,j,P i,j 1 }; e k and ê k are the errors for the unconstrained and the constrained scheme, respectively In an analogous way, t k and ˆt k are the CPU times for the unconstrained and the constrained scheme The empirical order is calculated as log e k 1 log e k ) ), )/ log Nk log N k 1 ), where ek and N k denote the quadratic error and number of points per side, which are calculated for a given grid and the coarser one for the same region The examination of the empirical errors shows that both schemes produce numerical solutions which are very close, despite one of them, the heuristic, was defined by an unconstrained problem However, the algebraic systems assembled for the heuristic scheme are solved faster in all the tests Another important issue is that the presence of strong non convexities in the boundaries causes the generation of elongated grids cells which affect the accuracy of the numerical solution for both schemes, although the unconstrained heuristic scheme turns out to be more robust than the constrained one in most of the numerical tests and always more robust in the case of the refined grids of the test set a) Nevertheless, in a similar way to the case of finite elements on very elongated triangles, convergence is expected to be achieved [17]

10 680 F Domínguez-Mota et al Grid N k e k Order ê k Order t k ˆt k E E ENG E E E E E E HAB E E E E E E MIC E E E E E E UCH E E E E Table 1: Problem 1 data Test set a Grid N k e k Order ê k Order t k ˆt k E E ENG E E E E E E HAB E E E E E E MIC E E E E E E UCH E E E E Table 2: Problem 2 data Test set a

11 An heuristic finite difference scheme 681 Grid N k e k Order ê k Order t k ˆt k E E ENG E E E E E E HAB E E E E E E MIC E E E E E E UCH E E E E Table 3: Problem 1 data Test set b Grid N k e k Order ê k Order t k ˆt k E E ENG E E E E E E HAB E E E E E E MIC E E E E E E UCH E E E E Table 4: Problem 2 data Test set b

12 682 F Domínguez-Mota et al 4 Conclusion As follows from the numerical tests, the proposed heuristic scheme produces satisfactory results and, even though it might be in theory slightly less accurate than the scheme addressed in [8], it was as robust as the scheme defined by a local constrained optimization problem 4 In addition, in all the tests, it was notably faster due to the lack of constraints This leds to conclude that it can be used with ease in order to get reliable approximations to the solution of linear second-order partial equations in irregular regions Besides, its implementation can be made in a very simple direct way Acknowledgements We want to thank CIC-UMSNH 916, and Grant SEP-PROMEP Aplicaciones de la optimización numérica a la solución de diversos problemas de cómputo científico for the financial support for this work References [1] A Chávez-González, A Cortés-Medina and JG Tinoco-Ruiz, A direct finite-difference scheme for solving PDEs over general two-dimensional regions, Applied Numerical Mathematics, ), [2] JE Castillo and RD Grone, A matrix analysis approach to higher-order approximations for divergence and gradients satisfying a global conservation law, SIAM J Matrix Anal Appl, Vol 25, No ), [3] P Barrera-Sánchez, L Castellanos, FJ Domínguez-Mota, GF González- Flores and APérez-Domínguez, Adaptive discrete harmonic grid generation, Math Comput Simulation, ), [4] UNAMALLA, An Automatic Package for Numerical Grid Generation, [5] M Shashkov, Conservative finite difference methods on general grids, Symbolic and Numeric Computation Series, CRC Press, 1996) [6] RB Kellogg, Difference equations for the neutron diffusion equations in hexagonal geometry, Contract AT-11-1-GEN-14 [7] RB Kellogg, Difference equations on a Mesh Arising from a General Triangulation, Mathematics of Computation, Vol 18, No ),

13 An heuristic finite difference scheme 683 [8] FJ Domínguez-Mota, S Mendoza-Armenta and JG Tinoco-Ruiz, Finite Difference Schemes Satisfying an Optimality Condition, MASCOT10 Proceedings, 2011) [9] JE Castillo, JM Hyman, M Shashkov and S Steinberg, Fourth- and sixth-order conservative finite difference approximations of the divergence and gradient, Applied Numerical Mathematics, ), [10] P Knupp and S Steinberg, Fundamentals of Grid Generation, CRC Press, Boca Raton, FL, 1994) [11] P Barrera-Sánchez, FJ Domínguez-Mota, GF González-Flores and JG Tinoco-Ruiz, Generating quality structured convex grids on irregular regions, Electronic Transactions on Numerical Analysis, ), [12] SA Ivanenko, Harmonic mappings, in Handbook of Grid Generation, CRC Press, Boca Raton, FL, 1999), [13] JE Castillo, ed, Mathematical Aspects of Numerical Grid Generation, Frontiers Appl Math 8, SIAM, Philadelphia, 1991) [14] JE Castillo, Mimetic discretization methods, Routledge Chapman & Hal, 2013) [15] M Celia and W Gray, Numerical Methods for Differential Equations, Prentice-Hall, 1992) [16] JG Tinoco-Ruiz and P Barrera-Sánchez, Smooth and convex grid generation over general plane regions, Math Comput Simulation, ), [17] A Hannukainen and S Korotov, The maximum angle condition is not necessary for convergence of the finite element method, Numerische Mathematik, ), [18] S Steinberg and P J Roache, Variational grid generation, Numer Methods Partial Differential Equations, ), Received: December 7, 2013