{"title":"Numerical Study of Iterative Methods for the Solution of the Dirichlet-Neumann Map for Linear Elliptic PDEs on Regular Polygon Domains","authors":"A. G. Sifalakis, E. P. Papadopoulou, Y. G. Saridakis","volume":9,"journal":"International Journal of Mathematical and Computational Sciences","pagesStart":410,"pagesEnd":416,"ISSN":"1307-6892","URL":"https:\/\/publications.waset.org\/pdf\/14250","abstract":"A generalized Dirichlet to Neumann map is\none of the main aspects characterizing a recently introduced\nmethod for analyzing linear elliptic PDEs, through which it\nbecame possible to couple known and unknown components\nof the solution on the boundary of the domain without\nsolving on its interior. For its numerical solution, a well conditioned\nquadratically convergent sine-Collocation method\nwas developed, which yielded a linear system of equations\nwith the diagonal blocks of its associated coefficient matrix\nbeing point diagonal. This structural property, among others,\ninitiated interest for the employment of iterative methods for\nits solution. In this work we present a conclusive numerical\nstudy for the behavior of classical (Jacobi and Gauss-Seidel)\nand Krylov subspace (GMRES and Bi-CGSTAB) iterative\nmethods when they are applied for the solution of the Dirichlet\nto Neumann map associated with the Laplace-s equation\non regular polygons with the same boundary conditions on\nall edges.","references":"[1] A.S.Fokas, A unified transform method for solving linear and\ncertain nonlinear PDEs, Proc. R. Soc. London A53 (1997),\n1411-1443.\n[2] S. Fulton, A.S. Fokas and C. Xenophontos, An Analytical\nMethod for Linear Elliptic PDEs and its Numerical Implementation,\nJ. of CAM 167 (2004), 465-483.\n[3] A. Sifalakis, A.S. Fokas, S. Fulton and Y.G. Saridakis, The\nGeneralized Dirichlet-Neumann Map for Linear Elliptic PDEs\nand its Numerical Implementation, J. of Comput. and Appl.\nMaths. (in press)\n[4] A.S.Fokas, Two-dimensional linear PDEs in a convex polygon,\nProc. R. Soc. London A 457 (2001), 371-393.\n[5] A.S. Fokas, A New Transform Method for Evolution PDEs,\nIMA J. Appl. Math. 67 (2002), 559.\n[6] G. Dassios and A.S. Fokas, The Basic Elliptic Equations in\nan Equilateral Triangle, Proc. R. Soc. Lond. A 461 (2005),\n2721-2748.\n[7] Y. Saad and M. Schultz, GMRES: a generalized minimal\nresidual algorithm for solving nonsymmetric linear systems,\nSIAM J. Sci. Statist. Comput., 7,1986,pp. 856-869.\n[8] H.A. Van Der Vorst, Bi-CGSTAB: A fast and smoothly\nconverging variant of Bi-CG for the solution of nonsymmetric\nlinear systems, SIAM J. Sci. Statist. Comput., 13,1992, pp.\n631-644.","publisher":"World Academy of Science, Engineering and Technology","index":"Open Science Index 9, 2007"}