Numerical Study of Iterative Methods for the Solution of the Dirichlet-Neumann Map for Linear Elliptic PDEs on Regular Polygon Domains

A generalized Dirichlet to Neumann map is one of the main aspects characterizing a recently introduced method for analyzing linear elliptic PDEs, through which it became possible to couple known and unknown components of the solution on the boundary of the domain without solving on its interior. For its numerical solution, a well conditioned quadratically convergent sine-Collocation method was developed, which yielded a linear system of equations with the diagonal blocks of its associated coefficient matrix being point diagonal. This structural property, among others, initiated interest for the employment of iterative methods for its solution. In this work we present a conclusive numerical study for the behavior of classical (Jacobi and Gauss-Seidel) and Krylov subspace (GMRES and Bi-CGSTAB) iterative methods when they are applied for the solution of the Dirichlet to Neumann map associated with the Laplace-s equation on regular polygons with the same boundary conditions on all edges.

Authors:

• A. G. Sifalakis
• E. P. Papadopoulou
• Y. G. Saridakis

Keywords:

• Elliptic PDEs
• Dirichlet to Neumann Map
• Global Relation
• Collocation
• Iterative Methods
• Jacobi
• Gauss-Seidel
• GMRES
• Bi-CGSTAB.

References:

[1] A.S.Fokas, A unified transform method for solving linear and
certain nonlinear PDEs, Proc. R. Soc. London A53 (1997),
1411-1443.
[2] S. Fulton, A.S. Fokas and C. Xenophontos, An Analytical
Method for Linear Elliptic PDEs and its Numerical Implementation,
J. of CAM 167 (2004), 465-483.
[3] A. Sifalakis, A.S. Fokas, S. Fulton and Y.G. Saridakis, The
Generalized Dirichlet-Neumann Map for Linear Elliptic PDEs
and its Numerical Implementation, J. of Comput. and Appl.
Maths. (in press)
[4] A.S.Fokas, Two-dimensional linear PDEs in a convex polygon,
Proc. R. Soc. London A 457 (2001), 371-393.
[5] A.S. Fokas, A New Transform Method for Evolution PDEs,
IMA J. Appl. Math. 67 (2002), 559.
[6] G. Dassios and A.S. Fokas, The Basic Elliptic Equations in
an Equilateral Triangle, Proc. R. Soc. Lond. A 461 (2005),
2721-2748.
[7] Y. Saad and M. Schultz, GMRES: a generalized minimal
residual algorithm for solving nonsymmetric linear systems,
SIAM J. Sci. Statist. Comput., 7,1986,pp. 856-869.
[8] H.A. Van Der Vorst, Bi-CGSTAB: A fast and smoothly
converging variant of Bi-CG for the solution of nonsymmetric
linear systems, SIAM J. Sci. Statist. Comput., 13,1992, pp.
631-644.