Semicirculant Solvers and Boundary Corrections for First-Order Partial Differential Equations

Abstract

Fast solvers for systems of partial differential equations (PDEs) in two space dimensions are considered. The solvers are used as direct solution methods or as preconditioners for a conjugate gradient (CG)-like iteration. Mainly first-order PDEs are considered, but second-order terms may be included. Employing a semi-circulant framework, PDEs with constant coefficients in one space direction and arbitrary boundary conditions are considered. A factorization of the inverse of the difference approximation matrix is described. This factorization is exploited to derive a direct solver, where the complexity for the first right-hand side is $\mathcal{O}(n^{{3 / 2}} \log _2 n)$, but only $\mathcal{O}(n\log _2 n)$ for subsequent right-hand sides. From the factorization an iteration resembling the Schur complement matrix method known from domain decomposition is also constructed. The eigendecomposition of the iteration matrix is investigated. The new solution methods are compared with iterations with semicirculant preconditioners and to Gaussian elimination. An application solving the time-dependent, almost incompressible Navier–Stokes equations is also studied.