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

Iain Smears

TL;DR

This work develops a fully robust and efficient preconditioning strategy for solving parabolic inf-sup theory systems that requires the solution of a large nonsymmetric system at each time-step.

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.

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

TL;DR

This work develops a fully robust and efficient preconditioning strategy for solving parabolic inf-sup theory systems that requires the solution of a large nonsymmetric system at each time-step.

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.

Paper Structure

This paper contains 19 sections, 7 theorems, 84 equations, 2 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Approximation space
  4. The DG time-stepping method
  5. Reconstruction operator
  6. Negative norms and a result of Pearson and Wathen
  7. Preconditioners
  8. Left preconditioner
  9. Spectrally equivalent norm preconditioner
  10. Definition of the basis
  11. Construction of the norm preconditioner
  12. Implementation
  13. Computation of eigenfunctions and eigenvalues
  14. Implementation of preconditioners
  15. Numerical experiments
  16. ...and 4 more sections

Key Result

lemma 1

Let $A$ and $M$ be arbitrary symmetric positive definite matrices and let $\mu\geq 0$ be a nonnegative real number. Then we have

Figures (2)

  • Figure 1: Eigenfunctions $\{\varphi_j\}_{j=0}^p$ defined by \ref{['eq:eigenfunctions']} and \ref{['eq:eigenfunctions_normalized']}, computed for $p=4$ by the method of section \ref{['sec:comp_eigenfunctions']} and ordered by decreasing eigenvalue $\lambda_j\geq \lambda_{j+1}$. Note that the $j$-th eigenfunction $\varphi_j$ need not be of degree at most $j$. Furthermore, the set of eigenfunctions generally depends on $p$, although they are independent of $A$, $M$ and $\tau$.
  • Figure 2: Eigenvalues $\{\lambda_j\}_{j=0}^p$ computed by the method of section \ref{['sec:comp_eigenfunctions']} and arranged in decreasing order, for $p=5$, $10$, $100$ and $100$. All plots are on a logarithmic scale, except for $p=5$ which is given on a semilogarithmic scale. It appears that the bounds of Theorem \ref{['thm:eigenvalue_distribution']} are sharp.

Theorems & Definitions (17)

  • remark 1
  • remark 2
  • remark 3
  • lemma 1
  • proof
  • theorem 1
  • proof
  • remark 4
  • theorem 2
  • theorem 3
  • ...and 7 more