Robust and efficient preconditioners for the discontinuous Galerkin time-stepping method

TL;DR

This paper introduces a robust preconditioning strategy for the discontinuous Galerkin time-stepping method, significantly improving the efficiency and stability of solving large parabolic PDE systems.

Contribution

It develops a novel preconditioning approach based on parabolic inf-sup theory that guarantees low condition numbers regardless of problem parameters.

Findings

Condition number bounded by 4 for all parameters

Fast convergence demonstrated in numerical experiments

Feasibility of high-order large-scale problem solutions

Abstract

The discontinuous Galerkin time-stepping method has many advantageous properties for solving parabolic equations. However, it requires the solution of a large nonsymmetric system at each time-step. This work develops a fully robust and efficient preconditioning strategy for solving these systems. Drawing on parabolic inf-sup theory, we first construct a left preconditioner that transforms the linear system to a symmetric positive definite problem to be solved by the preconditioned conjugate gradient algorithm. We then prove that the transformed system can be further preconditioned by an ideal block diagonal preconditioner, leading to a condition number bounded by 4 for any time-step size, any approximation order and any positive-definite self-adjoint spatial operators. Numerical experiments demonstrate the low condition numbers and fast convergence of the algorithm for both ideal and approximate preconditioners, and show the feasibility of the high-order solution of large problems.