Robust PRESB preconditioning of a 3-dimensional space-time finite element method for parabolic problems. (English) Zbl 1541.65096
Summary: We present a recently developed preconditioning of square block matrices (PRESB) to be used within a parallel method to solve linear systems of equations arising from tensor-product discretizations of initial boundary-value problems for parabolic partial differential equations. We consider weak formulations in Bochner-Sobolev spaces and tensor-product finite element approximations for the heat and eddy current equations. The fast diagonalization method is employed to decouple the arising linear system of equations into auxiliary spatial complex-valued linear systems that can be solved concurrently. We prove that the real part of the system matrix is positive definite, which allows us to accelerate the flexible generalized minimal residual method (FGMRES) by the PRESB preconditioner. The action of PRESB on a vector includes two solutions with positive definite matrices. The spectrum of the preconditioned system lies between 1/2 and 1. Finally, we combine the PRESB-FGMRES method with multigrid-CG iterations and illustrate the numerical efficiency and the robustness for spatial discretizations up to 12 millions degrees of freedom.
MSC:
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 65M22 | Numerical solution of discretized equations for initial value and initial-boundary value problems involving PDEs |
| 65M55 | Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs |
| 65F08 | Preconditioners for iterative methods |
Keywords:
parabolic problems; parallel space-time methods; finite element methods; fast diagonalization method; PRESB preconditioning; multigridReferences:
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