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

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• 作者：A.G. Sifalakis ; S.R. Fulton ; E.P. Papadopoulou ; Y.G. Saridakis
• 关键词：Elliptic PDEs ; Dirichlet– ; Neumann map ; Global relation ; Collocation ; Iterative methods ; Jacobi ; Gauss– ; Seidel ; GMRES ; Bi-CGSTAB
• 刊名：Journal of Computational and Applied Mathematics
• 出版年：2009
• 出版时间：1 May 2009
• 年：2009
• 卷：227
• 期：1
• 页码：171-184
• 全文大小：1049 K

文摘

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.