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

Authors
Journal
Journal of Computational and Applied Mathematics
0377-0427
Publisher
Elsevier
Volume
236
Issue
9
Identifiers
DOI: 10.1016/j.cam.2011.12.011
Keywords
  • Elliptic Pdes
  • Dirichlet–Neumann Map
  • Global Relation
  • Collocation
  • Block Circulant Matrices
  • Fast Fourier Transform (Fft)
Disciplines
  • Computer Science

Abstract

Abstract 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.

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