A direct solver for variable coefficient elliptic PDEs discretized via a composite spectral collocation method

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(N1.5)O(N1.5) complexity for the pre-computation and O(NlogN)O(NlogN) 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 100×100100×100 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.