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Interior Penalties for Summation-by-Parts Discretizations of Linear Second-Order Differential Equations

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Abstract

This work focuses on simultaneous approximation terms (SATs) for multidimensional summation-by-parts (SBP) discretizations of linear second-order partial differential equations with variable coefficients. Through the analysis of adjoint consistency and stability, we present several conditions on the SAT penalties for general operators, including those operators that do not have nodes on their boundary or do not correspond with a collocation discontinuous Galerkin method. Based on these conditions, we generalize the modified scheme of Bassi and Rebay and the symmetric interior penalty Galerkin method to SBP-SAT discretizations. Numerical experiments are carried out on unstructured grids with triangular elements to verify the theoretical results.

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Notes

  1. Equivalently, we can consider the generalized eigenvalue problem \(\mathsf {A} \varvec{v}_{i,j} = \mu _{i,j}\mathsf {H} \varvec{v}_{i,j}\); see, for example, [38, Chapter 8]

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Acknowledgements

Funding was provided by National Science Foundation (Grant No. 1554253) and Air Force Office of Scientific Research (Grant No.FA9550-15-1-0242)

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  1. Rensselaer Polytechnic Institute, Troy, NY, USA

    Jianfeng Yan, Jared Crean & Jason E. Hicken

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  1. Jianfeng Yan
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Correspondence to Jianfeng Yan.

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Yan, J., Crean, J. & Hicken, J.E. Interior Penalties for Summation-by-Parts Discretizations of Linear Second-Order Differential Equations. J Sci Comput 75, 1385–1414 (2018). https://doi.org/10.1007/s10915-017-0591-8

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