Efficient preconditioning of variational time discretization methods for parabolic Partial Differential Equations

This paper is concerned with the design of efficient preconditioners for systems arising from variational time discretization methods for parabolic partial differential equations. We consider the first order discontinuous Galerkin method (dG(1)) and the second order continuous Galerkin Petrov method (cGP(2)). The time-discrete formulation of these methods leads to a coupled 2 × 2 block system whose efficient solution strongly depends on efficient preconditioning strategies. The preconditioner proposed in this paper is based on a Schur complement formulation for the so called essential unknown. By introducing an inexact factorization of this ill-conditioned fourth order operator, we are able to circumvent complex arithmetic and prove uniform bounds for the condition number of the preconditioned system. In addition, the resulting preconditioned operator is symmetric and positive definite, therefore allowing for the usage of efficient Krylov subspace solvers such as the conjugate gradient method. For both the dG(1) and cGP(2) method, we provide optimal choices for the sole parameter of the preconditioner and deduce corresponding upper bounds for the condition number of the resulting preconditioned system. Several numerical experiments including the heat equation and a convection-diffusion example confirm the theoretical findings