Fast Structured Direct Spectral Methods for Differential Equations with Variable Coefficients, I. The One-Dimensional Case

We study the rank structures of the matrices in Fourierand Chebyshev-spectral methods for differential equations with variable coefficients in one dimension. We show analytically that these matrices have a so-called low-rank property, not only for constant or smooth variable coefficients, but also for coefficients with steep gradients and/or high variations (large ratios in their maximum-minimum function values). We develop a matrix-free direct spectral solver, which uses only a small number of matrix-vector products to construct a structured approximation to the original discretized matrix A, without the need to explicitly form A. This is followed by fast structured matrix factorizations and solutions. The overall direct spectral solver has O(N log N) complexity and O(N) memory requirement. Numerical tests for several important but notoriously difficult problems show the superior efficiency and accuracy of our direct spectral solver, especially when iterative methods have severe difficulties in the convergence.