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A direct solver for variable coefficient elliptic PDEs discretized via a composite spectral collocation method
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DOI：10.1016/j.jcp.2013.02.019.png)
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Abstract
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A numerical method for variable coefficient elliptic problems on two-dimensional domains is presented. The method is based on high-order spectral approximations and is designed for problems with smooth solutions. The resulting system of linear equations is solved using a direct solver with O(N-1.5) complexity for the pre-computation and O(N logN) complexity for the solve. The fact that the solver is direct is a principal feature of the scheme, and makes it particularly well suited to solving problems for which iterative solvers struggle; in particular for problems with highly oscillatory solutions. Numerical examples demonstrate that the scheme is fast and highly accurate. For instance, using a discretization with 12 points per wavelength, a Helmholtz problem on a domain of size 100 x 100 wavelengths was solved to ten correct digits. The computation was executed on a standard laptop; it involved 1.6 M degrees of freedom and required 100 s for the pre-computation, and 0.3 s for the actual solve. (C) 2013 Elsevier Inc. All rights reserved.
Keywords:
Helmholtz equation
Spectral method
Direct solver
Fast direct solver
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