Efficient numerical solution of the generalized Dirichlet–Neumann map for linear elliptic PDEs in regular polygon domains
| Autor: | A. G. Sifalakis, E. P. Papadopoulou, Y. G. Saridakis |
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| Rok vydání: | 2012 |
| Předmět: |
Fast Fourier Transform (FFT)
Collocation Greek mathematics mathematics greek greek mathematics Applied Mathematics Linear system Mathematical analysis MathematicsofComputing_NUMERICALANALYSIS Regular polygon Boundary (topology) Basis function Elliptic PDEs Matrix (mathematics) Computational Mathematics Dirichlet–Neumann map Block circulant matrices Boundary value problem Global Relation Coefficient matrix Mathematics |
| Zdroj: | Journal of Computational and Applied Mathematics. 236(9):2515-2528 |
| ISSN: | 0377-0427 |
| DOI: | 10.1016/j.cam.2011.12.011 |
| Popis: | Summarization: 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. Presented on: Journal of Computational and Applied Mathematics |
| Databáze: | OpenAIRE |
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