Direct and iterative solution of the generalized Dirichlet-Neumann map for elliptic PDEs on square domains

Authors:
A. G. Sifalakis;S. R. Fulton;E. P. Papadopoulou;Y. G. Saridakis
Affiliations:
Applied Mathematics and Computers Lab, Department of Sciences, Technical University of Crete, 73100 Chania, Greece;Department of Mathematics and Computer Science, Clarkson University, Potsdam NY 13699-5815, USA;Applied Mathematics and Computers Lab, Department of Sciences, Technical University of Crete, 73100 Chania, Greece;Applied Mathematics and Computers Lab, Department of Sciences, Technical University of Crete, 73100 Chania, Greece
Venue:
Journal of Computational and Applied Mathematics
Year:
2009

Citing 5
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Abstract

In this work we derive the structural properties of the Collocation coefficient matrix associated with the Dirichlet-Neumann map for Laplace's equation on a square domain. The analysis is independent of the choice of basis functions and includes the case involving the same type of boundary conditions on all sides, as well as the case where different boundary conditions are used on each side of the square domain. Taking advantage of said properties, we present efficient implementations of direct factorization and iterative methods, including classical SOR-type and Krylov subspace (Bi-CGSTAB and GMRES) methods appropriately preconditioned, for both Sine and Chebyshev basis functions. Numerical experimentation, to verify our results, is also included.