Interpolatory HDG Method for Parabolic Semilinear PDEs
For researchers solving nonlinear parabolic PDEs, this method offers a more efficient alternative to standard HDG without sacrificing accuracy, though it is an incremental improvement.
The paper introduces an interpolatory HDG method for parabolic semilinear PDEs that replaces numerical quadrature with an interpolation procedure, reducing computational cost while maintaining convergence order. Numerical experiments in 2D and 3D demonstrate its efficiency.
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.