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.
B. Cockburn et al., "Interpolatory HDG Method for Parabolic Semilinear PDEs," Journal of Scientific Computing, vol. 79, no. 3, pp. 1777-1800, Springer Verlag, Jun 2019.
The definitive version is available at https://doi.org/10.1007/s10915-019-00911-8
Mathematics and Statistics
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)
Article - Journal
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01 Jun 2019