Efficient numerical solution of the generalized Dirichlet-Neumann map for linear elliptic PDEs in regular polygon domains

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

卷:236

Article

Saridakis, Y. G.¹  Sifalakis, A. G.¹  Papadopoulou, E. P.¹ 

[1] Tech Univ Crete, Dept Sci, Appl Math & Comp Lab, Khania 73100, Greece

关键词: Elliptic PDEs;    Dirichlet-Neumann map;    Global Relation;    Collocation;    Block circulant matrices;    Fast Fourier Transform (FFT);   

DOI  :  10.1016/j.cam.2011.12.011

来源: Elsevier

【 摘 要 】

A new and novel approach for analyzing boundary value problems for linear and for integrable nonlinear PDEs was recently introduced. For linear elliptic PDEs, an important aspect of this approach is the characterization of a generalized Dirichlet-Neumann map: given the derivative of the solution along a direction of an arbitrary angle to the boundary, the derivative of the solution perpendicularly to this direction is computed without solving on the interior of the domain. For this computation, a collocation-type numerical method has been recently developed. Here, we study the collocation's coefficient matrix properties. We prove that, for the Laplace's equation on regular-polygon domains with the same type of boundary conditions on each side, the collocation matrix is block circulant, independently of the choice of basis functions. This leads to the deployment of the FFT for the solution of the associated collocation linear system, yielding significant computational savings. Numerical experiments are included to demonstrate the efficiency of the whole computation. (C) 2012 Elsevier B.V. All rights reserved.

【 授权许可】

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