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A discretization scheme for variable coefficient Helmholtz problems ontwo-dimensional domains is presented. The scheme is based on high-orderspectral approximations and is designed for problems with smooth solutions. Theresulting system of linear equations is solved using a direct solver withO(N^1.5) complexity for the pre-computation and O(N log N) complexity for thesolve. The fact that the solver is direct is a principal feature of the scheme,since iterative methods tend to struggle with the Helmholtz equation. Numericalexamples demonstrate that the scheme is fast and highly accurate. For instance,using a discretization with 12 points per wave-length, a Helmholtz problem on adomain of size 100 x 100 wavelengths was solved to ten correct digits. Thecomputation was executed on an office desktop; it involved 1.6M degrees offreedom and required 100 seconds for the pre-computation, and 0.3 seconds forthe actual solve.

A composite spectral scheme for variable coefficient Helmholtz problems