scholarly journals Unified Hybridization of Discontinuous Galerkin, Mixed, and Continuous Galerkin Methods for Second Order Elliptic Problems

2009 ◽  
Vol 47 (2) ◽  
pp. 1319-1365 ◽  
Author(s):  
Bernardo Cockburn ◽  
Jayadeep Gopalakrishnan ◽  
Raytcho Lazarov
Keyword(s):  
Discontinuous Galerkin ◽  
Elliptic Problems ◽  
Second Order ◽  
Galerkin Methods ◽  
Continuous Galerkin ◽  
Continuous Galerkin Methods ◽  
Second Order Elliptic Problems

Families of interior penalty hybridizable discontinuous Galerkin methods for second order elliptic problems

2020 ◽  
Vol 28 (3) ◽  
pp. 161-174
Author(s):  
Maurice S. Fabien ◽  
Matthew G. Knepley ◽  
Beatrice M. Riviere
Keyword(s):  
Error Estimates ◽  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Methods ◽  
Computational Cost ◽  
Elliptic Problems ◽  
Second Order ◽  
Three Dimensions ◽  
Galerkin Methods ◽  
Interior Penalty ◽  
Second Order Elliptic Problems

AbstractThe focus of this paper is the analysis of families of hybridizable interior penalty discontinuous Galerkin methods for second order elliptic problems. We derive a priori error estimates in the energy norm that are optimal with respect to the mesh size. Suboptimal L2-norm error estimates are proven. These results are valid in two and three dimensions. Numerical results support our theoretical findings, and we illustrate the computational cost of the method.

scholarly journals Analysis of the SBP-SAT Stabilization for Finite Element Methods Part I: Linear Problems

2020 ◽  
Vol 85 (2) ◽  
Author(s):  
R. Abgrall ◽  
J. Nordström ◽  
P. Öffner ◽  
S. Tokareva
Keyword(s):  
Finite Element ◽  
Boundary Conditions ◽  
Discontinuous Galerkin ◽  
Finite Element Methods ◽  
Main Idea ◽  
Galerkin Methods ◽  
Exact Integration ◽  
Galerkin Scheme ◽  
Continuous Galerkin ◽  
Continuous Galerkin Methods

AbstractIn the hyperbolic community, discontinuous Galerkin (DG) approaches are mainly applied when finite element methods are considered. As the name suggested, the DG framework allows a discontinuity at the element interfaces, which seems for many researchers a favorable property in case of hyperbolic balance laws. On the contrary, continuous Galerkin methods appear to be unsuitable for hyperbolic problems and there exists still the perception that continuous Galerkin methods are notoriously unstable. To remedy this issue, stabilization terms are usually added and various formulations can be found in the literature. However, this perception is not true and the stabilization terms are unnecessary, in general. In this paper, we deal with this problem, but present a different approach. We use the boundary conditions to stabilize the scheme following a procedure that are frequently used in the finite difference community. Here, the main idea is to impose the boundary conditions weakly and specific boundary operators are constructed such that they guarantee stability. This approach has already been used in the discontinuous Galerkin framework, but here we apply it with a continuous Galerkin scheme. No internal dissipation is needed even if unstructured grids are used. Further, we point out that we do not need exact integration, it suffices if the quadrature rule and the norm in the differential operator are the same, such that the summation-by-parts property is fulfilled meaning that a discrete Gauss Theorem is valid. This contradicts the perception in the hyperbolic community that stability issues for pure Galerkin scheme exist. In numerical simulations, we verify our theoretical analysis.

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