Abstract
The present paper discusses families of Interior Penalty Discontinuous Galerkin (IP) methods for solving heterogeneous and anisotropic diffusion problems. Specifically, we focus on distinctive schemes, namely the Hybridized-, Embedded-, and Weighted-IP schemes, leading to final matrixes of different sizes and sparsities. Both the Hybridized- and Embedded-IP schemes are eligible for static condensation, and their degrees of freedom are distributed on the mesh skeleton. In contrast, the unknowns are located inside the mesh elements for the Weighted-IP variant. For a given mesh, it is well-known that the number of degrees of freedom related to the standard Discontinuous Galerkin methods increases more rapidly than those of the skeletal approaches (Hybridized- and Embedded-IP). We then quantify the impact of the static condensation procedure on the computational performances of the different IP classes in terms of robustness, accuracy, and CPU time. To this aim, numerical experiments are investigated by considering strong heterogeneities and anisotropies. We analyze the fixed error tolerance versus the run time and mesh size to guide our performance criterion. We also outlined some relationships between these Interior Penalty schemes. Eventually, we confirm the superiority of the Hybridized- and Embedded-IP schemes, regardless of the mesh, the polynomial degree, and the physical properties (homogeneous, heterogeneous, and/or anisotropic).
Highlights
The Discontinuous Galerkin (DG) methods were firstly introduced by Reed and Hill (1973) for the neutron transport phenomenon
The final matrix system of DG methods leads to a larger stencil with a higher number of coupled degrees of freedom (DOFs), which is quite challenging for largescale problems (Rivière, 2008)
This section provided several numerical experiments, which aims to investigate the performances of the Embedded Interior Penalty method (EIP), Hybridized Interior Penalty method (HIP) and Weighted-Interior Penalty (WIP)
Summary
The Discontinuous Galerkin (DG) methods were firstly introduced by Reed and Hill (1973) for the neutron transport phenomenon Since their introduction, the DG methods have become a relevant class of finite element schemes for modeling physical processes. The final matrix system of DG methods leads to a larger stencil with a higher number of coupled degrees of freedom (DOFs), which is quite challenging for largescale problems (Rivière, 2008). Following these observations, Cockburn et al (2009b) introduced a new DG discretization scheme to overcome these drawbacks called the Hybridized Discontinuous Galerkin (HDG) methods
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