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Continuous/Discontinuous Galerkin Difference Discretizations of High-Order Differential Operators

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Abstract

We develop continuous/discontinuous discretizations for high-order differential operators using the Galerkin Difference approach. Grid dispersion analyses are performed that indicate a nodal superconvergence in the \(\ell ^2\) norm. A treatment of the boundary conditions is described that ultimately leads to moderate growth in the spectral radius of the operators with polynomial degree, and in general the norms of the Galerkin Difference differentiation operators are significantly smaller than those arising from standard elements. Lastly, we observe that with the use of the Galerkin Difference space, the standard penalty terms required for discretizing high-order operators are not needed. Numerical results confirm the conclusions of the analyses performed.

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Data Availability

The data generated during and/or analysed during the current study are available from the corresponding author upon reasonable request.

Notes

  1. We use the convention that \(\ell ^2\) denotes the discrete- \(L^2\) norm.

  2. In fact the ghost DoFs describe aspects of the solution on the domain interior, but it is conceptually useful to think of them as living outside the domain.

  3. We use the term “backward Euler-like” since at its core the low-order scheme is actually a discretization of a Picard integral formulation, although the resulting scheme may be identical to backward-Euler.

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Funding

J.W. Banks and B. Brett Buckner were supported in part by contracts from the U.S. Department of Energy ASCR Applied Math Program, and by a U.S. Presidential Early Career Award for Scientists and Engineers. T. Hagstrom was supported in part by NSF Grant DMS-2012296. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, NY, 12180, USA

    J. W. Banks & B. Brett Buckner

  2. Department of Mathematics, Southern Methodist University, PO Box 750156, Dallas, TX, 75275-0156, USA

    T. Hagstrom

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  1. J. W. Banks
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  2. B. Brett Buckner
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  3. T. Hagstrom
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Correspondence to J. W. Banks.

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Banks, J.W., Buckner, B.B. & Hagstrom, T. Continuous/Discontinuous Galerkin Difference Discretizations of High-Order Differential Operators. J Sci Comput 92, 45 (2022). https://doi.org/10.1007/s10915-022-01891-y

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  • DOI: https://doi.org/10.1007/s10915-022-01891-y

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