An Accurate Fourier-Spectral Solver for Variable Coefficient Elliptic Equations

We develop a solver for nonseparable, self adjoint elliptic equations with a variable coefficient. If the coefficient is the square of a harmonic function,a transformation of the dependent variable, results in a constant coefficient Poisson equation. A highly accurate, fast, Fourier-spectral algorithm can solve this equation. When the square root of the coefficient is not harmonic, we approximate it by a harmonic function. A small number of correction steps are then required to achieve high accuracy. The procedure is particularly efficient when the approximation error is small. For a given function this error becomes smaller as the size of the domain decreases. A highly parallelizable, hierarchical procedure allows a decomposition into small sub-domains. Numerical experiments illustrate the accuracy of the approach even at very coarse resolutions. Key-Words: Fast spectral direct solver, Poisson equation, nonseparable elliptic equations, correction steps.