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Journal of Scientific Computing

, Volume 79, Issue 3, pp 1777–1800 | Cite as

Interpolatory HDG Method for Parabolic Semilinear PDEs

  • Bernardo Cockburn
  • John R. SinglerEmail author
  • Yangwen Zhang
Article

Abstract

We propose the interpolatory hybridizable discontinuous Galerkin (Interpolatory HDG) method for a class of scalar parabolic semilinear PDEs. The Interpolatory HDG method uses an interpolation procedure to efficiently and accurately approximate the nonlinear term. This procedure avoids the numerical quadrature typically required for the assembly of the global matrix at each iteration in each time step, which is a computationally costly component of the standard HDG method for nonlinear PDEs. Furthermore, the Interpolatory HDG interpolation procedure yields simple explicit expressions for the nonlinear term and Jacobian matrix, which leads to a simple unified implementation for a variety of nonlinear PDEs. For a globally-Lipschitz nonlinearity, we prove that the Interpolatory HDG method does not result in a reduction of the order of convergence. We display 2D and 3D numerical experiments to demonstrate the performance of the method.

Keywords

Hybridizable discontinuous Galerkin method Interpolatory method Newton iteration 

Mathematics Subject Classification

65M60 65L12 

Notes

Acknowledgements

J. Singler and Y. Zhang were supported in part by National Science Foundation Grant DMS-1217122. J. Singler and Y. Zhang thank the IMA for funding research visits, during which some of this work was completed. Y. Zhang thanks Zhu Wang for many valuable conversations.

Compliance with Ethical Standards

Conflict of interest

The authors declare that they have no conflict of interest.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of MathematicsUniversity of MinnesotaMinneapolisUSA
  2. 2.Department of Mathematics and StatisticsMissouri University of Science and TechnologyRollaUSA
  3. 3.Department of Mathematical SciencesUniversity of DelawareNewarkUSA

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