Spectral domain embedding for elliptic PDEs in complex domains

Spectral methods are a class of methods for solving partial differential equations (PDEs). When the solution of the PDE is analytic, it is known that the spectral solutions converge exponentially as a function of the number of modes used. The basic spectral method works only for regular domains such...

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Bibliographic Details
Published in:Journal of computational and applied mathematics 2009-03, Vol.225 (2), p.541-557
Main Author: Lui, S.H.
Format: Article
Language:English
Subjects:
Domain embedding
Exact sciences and technology
Fictitious domain
Functions of a complex variable
Irregular domain
Mathematical analysis
Mathematics
Numerical analysis
Numerical analysis. Scientific computation
Partial differential equations
Partial differential equations, boundary value problems
Partial differential equations, initial value problems and time-dependant initial-boundary value problems
Pseudospectral method
Sciences and techniques of general use
Spectral collocation
Spectral method
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Summary:Spectral methods are a class of methods for solving partial differential equations (PDEs). When the solution of the PDE is analytic, it is known that the spectral solutions converge exponentially as a function of the number of modes used. The basic spectral method works only for regular domains such as rectangles or disks. Domain decomposition methods/spectral element methods extend the applicability of spectral methods to more complex geometries. An alternative is to embed the irregular domain into a regular one. This paper uses the spectral method with domain embedding to solve PDEs on complex geometry. The running time of the new algorithm has the same order as that for the usual spectral collocation method for PDEs on regular geometry. The algorithm is extremely simple and can handle Dirichlet, Neumann boundary conditions as well as nonlinear equations.
ISSN:0377-0427
1879-1778
DOI:10.1016/j.cam.2008.08.034