Matrix-free multigrid block-preconditioners for higher order Discontinuous Galerkin discretisations
| Autor: | Peter Bastian, Eike Hermann Müller, Marian Piatkowski, Steffen Müthing |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2019 |
| Předmět: |
G.4
Matrix-free methods Physics and Astronomy (miscellaneous) Computer science 65F08 65N30 65N55 65Y05 65Y10 65Y20 G.1.8 MathematicsofComputing_NUMERICALANALYSIS Multigrid Physics and Astronomy(all) Preconditioners Computational science Matrix (mathematics) Multigrid method Discontinuous Galerkin method Modelling and Simulation Discontinuous Galerkin FOS: Mathematics Mathematics - Numerical Analysis Elliptic PDE Sparse matrix D.1.3 Numerical Analysis DUNE Preconditioner Applied Mathematics Numerical Analysis (math.NA) Solver Computer Science Applications Computational Mathematics Elliptic partial differential equation Modeling and Simulation |
| Zdroj: | Bastian, P, Müller, E, Muething, S & Piatkowski, M 2019, ' Matrix-free multigrid block-preconditioners for higher order Discontinuous Galerkin discretisations ', Journal of Computational Physics, vol. 394, pp. 417-439 . https://doi.org/10.1016/j.jcp.2019.06.001 |
| DOI: | 10.1016/j.jcp.2019.06.001 |
| Popis: | Efficient and suitably preconditioned iterative solvers for elliptic partial differential equations (PDEs) of the convection-diffusion type are used in all fields of science and engineering. To achieve optimal performance, solvers have to exhibit high arithmetic intensity and need to exploit every form of parallelism available in modern manycore CPUs. The computationally most expensive components of the solver are the repeated applications of the linear operator and the preconditioner. For discretisations based on higher-order Discontinuous Galerkin methods, sum-factorisation results in a dramatic reduction of the computational complexity of the operator application while, at the same time, the matrix-free implementation can run at a significant fraction of the theoretical peak floating point performance. Multigrid methods for high order methods often rely on block-smoothers to reduce high-frequency error components within one grid cell. Traditionally, this requires the assembly and expensive dense matrix solve in each grid cell, which counteracts any improvements achieved in the fast matrix-free operator application. To overcome this issue, we present a new matrix-free implementation of block-smoothers. Inverting the block matrices iteratively avoids storage and factorisation of the matrix and makes it is possible to harness the full power of the CPU. We implemented a hybrid multigrid algorithm with matrix-free block-smoothers in the high order DG space combined with a low order coarse grid correction using algebraic multigrid where only low order components are explicitly assembled. The effectiveness of this approach is demonstrated by solving a set of representative elliptic PDEs of increasing complexity, including a convection dominated problem and the stationary SPE10 benchmark. 28 pages, 10 figures, 10 tables; accepted for publication in Journal of Computational Physics |
| Databáze: | OpenAIRE |
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