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Journal Articles IMA Journal of Numerical Analysis Year : 2016

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

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.
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Dates and versions

hal-01357497 , version 1 (29-08-2016)

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Iain Smears. Robust and efficient preconditioners for the discontinuous Galerkin time-stepping method. IMA Journal of Numerical Analysis, 2016, ⟨10.1093/imanum/drw050⟩. ⟨hal-01357497⟩
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