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Badrkhani, Vahid; Hiemstra, René R.; Mika, Michał; Schillinger, Dominik

A matrix-free macro-element variant of the hybridized discontinuous Galerkin method. (English) Zbl 07772318

Int. J. Numer. Methods Eng. 124, No. 20, 4427-4452 (2023).
Summary: We investigate a macro-element variant of the hybridized discontinuous Galerkin (HDG) method, using patches of standard simplicial elements that can have non-matching interfaces. Coupled via the HDG technique, our method enables local refinement by uniform simplicial subdivision of each macro-element. By enforcing one spatial discretization for all macro-elements, we arrive at local problems per macro-element that are embarrassingly parallel, yet well balanced. Therefore, our macro-element variant scales efficiently to n-node clusters and can be tailored to available hardware by adjusting the local problem size to the capacity of a single node, while still using moderate polynomial orders such as quadratics or cubics. Increasing the local problem size means simultaneously decreasing, in relative terms, the global problem size, hence effectively limiting the proliferation of degrees of freedom. The global problem is solved via a matrix-free iterative technique that also heavily relies on macro-element local operations. We investigate and discuss the advantages and limitations of the macro-element HDG method via an advection-diffusion model problem.
© 2023 The Authors. International Journal for Numerical Methods in Engineering published by John Wiley & Sons Ltd.

MSC:

65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
65Y05 Parallel numerical computation
76M10 Finite element methods applied to problems in fluid mechanics
35Q35 PDEs in connection with fluid mechanics

Keywords:

domain decomposition; hybridized discontinuous Galerkin method; load balancing; local adaptive refinement; macro-elements; matrix-free; scalability

Software:

PETSc
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References:

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