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arXiv · 2210.10415

$hp$-robust multigrid solver on locally refined meshes for FEM discretizations of symmetric elliptic PDEs

Michael Innerberger·Ani Miraçi·Dirk Praetorius·Julian Streitberger
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Abstract

In this work, we formulate and analyze a geometric multigrid method for the iterative solution of the discrete systems arising from the finite element discretization of symmetric second-order linear elliptic diffusion problems. We show that the iterative solver contracts the algebraic error robustly with respect to the polynomial degree $p \ge 1$ and the (local) mesh size $h$. We further prove that the built-in algebraic error estimator which comes with the solver is $hp$-robustly equivalent to the algebraic error. The application of the solver within the framework of adaptive finite element methods with quasi-optimal computational cost is outlined. Numerical experiments confirm the theoretical findings.

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Michael Innerberger, Ani Miraçi, Dirk Praetorius, Julian Streitberger. 2022-10-19. $hp$-robust multigrid solver on locally refined meshes for FEM discretizations of symmetric elliptic PDEs. https://doi.org/10.1051/m2an%2F2023104

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