Using cylindrical algebraic decomposition and local Fourier analysis to study numerical methods: two examples

TL;DR

This paper combines local Fourier analysis with cylindrical algebraic decomposition to analyze convergence rates and error estimates in numerical methods for PDEs, demonstrating the approach on two specific examples.

Contribution

It introduces a novel combination of symbolic computation and Fourier analysis to evaluate numerical method properties, applicable to various problems.

Findings

Computed convergence rate of a multigrid method

Compared approximation errors for different discretizations

Demonstrated the versatility of the combined approach

Abstract

Local Fourier analysis is a strong and well-established tool for analyzing the convergence of numerical methods for partial differential equations. The key idea of local Fourier analysis is to represent the occurring functions in terms of a Fourier series and to use this representation to study certain properties of the particular numerical method, like the convergence rate or an error estimate. In the process of applying a local Fourier analysis, it is typically necessary to determine the supremum of a more or less complicated term with respect to all frequencies and, potentially, other variables. The problem of computing such a supremum can be rewritten as a quantifier elimination problem, which can be solved with cylindrical algebraic decomposition, a well-known tool from symbolic computation. The combination of local Fourier analysis and cylindrical algebraic decomposition is a machinery that can be applied to a wide class of problems. In the present paper, we will discuss two examples. The first example is to compute the convergence rate of a multigrid method. As second example we will see that the machinery can also be used to do something rather different: We will compare approximation error estimates for different kinds of discretizations.