Abstract
This paper proposes a new algorithm for fast matrix-free evaluation of linear operators based on hybridizable discontinuous Galerkin discretizations with sum factorization, exemplified for the convection-diffusion equation on quadrilateral and hexahedral elements. The matrix-free scheme is based on a formulation of the method in terms of the primal variable and the trace. The proposed method is shown to be up to an order of magnitude faster than the traditionally considered matrix-based formulation in terms of the trace only, despite using more degrees of freedom. The impact of the choice of basis on the evaluation cost is discussed, showing that Lagrange polynomials with nodes co-located with the quadrature points are particularly efficient.
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Notes
- 1.
Note that we do not consider the narrow case where the mesh is Cartesian, c is axis-aligned and element-wise constant, and κ is constant, when the matrix for (q, u) is separable and can be expressed as a Kronecker product of 1D matrices with the inverse given by the fast diagonalization method [9].
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Acknowledgements
This work was supported by the German Research Foundation (DFG) under the project “High-order discontinuous Galerkin for the exa-scale” (ExaDG) within the priority program “Software for Exascale Computing” (SPPEXA), grant agreement no. KO5206/1-1, KR4661/2-1 and WA1521/18-1.
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Kronbichler, M., Kormann, K., Wall, W.A. (2019). Fast Matrix-Free Evaluation of Hybridizable Discontinuous Galerkin Operators. In: Radu, F., Kumar, K., Berre, I., Nordbotten, J., Pop, I. (eds) Numerical Mathematics and Advanced Applications ENUMATH 2017. ENUMATH 2017. Lecture Notes in Computational Science and Engineering, vol 126. Springer, Cham. https://doi.org/10.1007/978-3-319-96415-7_53
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