Wavelet-in-time multigrid-in-space preconditioning of parabolic evolution equations

Abstract : Two space-time variational formulations of linear parabolic evolution equations are considered, one is symmetric and elliptic on the trial space while the other is not. In each case, a space-time Petrov--Galerkin discretization using suitable tensor product trial and test functions leads to a large linear system of equations. The well-posedness of this system with respect to parabolic norms induces a canonical preconditioner for the algebraic equations that arise after a choice of basis. For the iterative resolution of this algebraic system with parallelization in the temporal direction we propose a sparse algebraic wavelet-in-time transformation on possibly nonuniform temporal meshes. This transformation approximately block-diagonalizes the preconditioner, and the individual spatial blocks can then be inverted for instance by standard spatial multigrid methods in parallel. The performance of the preconditioner is documented in a series of numerical experiments.
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Roman Andreev. Wavelet-in-time multigrid-in-space preconditioning of parabolic evolution equations. SIAM Journal on Scientific Computing, Society for Industrial and Applied Mathematics, 2016, 〈10.1137/140998639〉. 〈hal-01184494〉



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