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

Saridakis Ioannis, Papadopoulou Eleni, Sifalakis Anastasios

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URI: http://purl.tuc.gr/dl/dias/2370F327-A11D-4367-93AF-9AC4CFF811EA
Year 2012
Type of Item Peer-Reviewed Journal Publication
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Bibliographic Citation Y. G. Saridakis, A. Sifalakis , E. P. Papadopoulou, “Efficient numerical solution of the generalized dirichlet-neumann map for linear elliptic PDEs on regular polygon domains," J. Comp. & Applied Math.,vol. 236 ,no. 9, pp. 2515-2528, 2012. doi:10.1016/j.cam.2011.12.011 https://doi.org/10.1016/j.cam.2011.12.011
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Summary

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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