Interpolatory HDG Method for Parabolic Semilinear PDEs


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.


Mathematics and Statistics

Research Center/Lab(s)

Center for High Performance Computing Research


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.

Keywords and Phrases

Galerkin methods; Interpolation; Jacobian matrices; Numerical methods; Discontinuous galerkin; Discontinuous Galerkin methods; Interpolatory; Lipschitz nonlinearities; Newton iterations; Numerical experiments; Numerical quadrature; Order of convergence; Iterative methods; Hybridizable discontinuous Galerkin method; Interpolatory method

International Standard Serial Number (ISSN)

0885-7474; 1573-7691

Document Type

Article - Journal

Document Version


File Type





© 2019 Springer Verlag, All rights reserved.

Publication Date

01 Jun 2019