Algorithms for massively parallel generic hp-adaptive finite element methods

Fehling, Marc

Book/Dissertation / PhD Thesis

FZJ-2020-02694

Algorithms for massively parallel generic hp-adaptive finite element methods

Fehling, M. (Corresponding author)FZJ*

2020
Forschungszentrum Jülich GmbH Zentralbibliothek, Verlag Jülich
ISBN: 978-3-95806-486-7

Jülich : Forschungszentrum Jülich GmbH Zentralbibliothek, Verlag, Schriften des Forschungszentrums Jülich. IAS Series 43, vii, 78 S. (2020) = Universität Wuppertal, 2020

Please use a persistent id in citations: http://hdl.handle.net/2128/25427 urn:nbn:de:0001-2020071402

Abstract: Efficient algorithms for the numerical solution of partial differential equations are required to solve problems on an economically viable timescale. In general, this is achieved by adapting the resolution of the discretization to the investigated problem, as well as exploiting hardware specifications. For the latter category, parallelization plays a major role for modern multi-core and multi-node architectures, especially in the context of high-performance computing. Using finite element methods, solutions are approximated by discretizing the function space of the problem with piecewise polynomials. With $\textit{hp}$-adaptive methods, the polynomial degrees of these basis functions may vary on locally refined meshes. We present algorithms and data structures required for generic hp-adaptive finite element software applicable for both continuous and discontinuous Galerkin methods on distributed memory systems. Both function space and mesh may be adapted dynamically during the solution process. We cover details concerning the unique enumeration of degrees of freedom with continuous Galerkin methods, the communication of variable size data, and load balancing. Furthermore, we present strategies to determine the type of adaptation based on error estimation and prediction as well as smoothness estimation via the decay rate of coefficients of Fourier and Legendre series expansions. Both refinement and coarsening are considered. A reference implementation in the open-source library deal. II$^{1}$ is provided and applied to the Laplace problem on a domain with a reentrant corner which invokes a singularity. With this example, we demonstrate the benefits of the $\textit{hp}$-adaptive methods in terms of error convergence and show that our algorithm scales up to 49,152 MPI processes.

Note: Universität Wuppertal, 2020

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