A local Fourier convergence analysis of a multigrid method using symbolic computation

For iterative solvers, besides convergence proofs that may state qualitative results for some classes of problems, straight-forward methods to compute (bounds for) convergence rates are of particular interest. A widelyused straight-forward method to analyze the convergence of numerical methods for solving discretized systems of partial differential equations (PDEs) is local Fourier analysis (or local mode analysis). The rates that can be computed with local Fourier analysis are typically the supremum of some rational function. In the past this supremum was merely approximated numerically by interpolation. We are interested in resolving the supremum exactly using a standard tool from symbolic computation: cylindrical algebraic decomposition (CAD). In this paper we work out the details of this symbolic local Fourier analysis for a multigrid solver applied to a PDE-constrained optimization problem.