scholarly journals Lower-order equivalent nonstandard finite element methods for biharmonic plates

2021 ◽  
Author(s):  
Neela Nataraj ◽  
Carsten Carstensen
Keyword(s):  
Finite Element ◽  
Discontinuous Galerkin ◽  
Finite Element Methods ◽  
Biharmonic Equation ◽  
Input Function ◽  
Comparison Results ◽  
Sobolev Norms ◽  
Discrete Norm ◽  
Numer Anal ◽  
Nonconforming Methods

The popular (piecewise) quadratic schemes for the biharmonic equation based on triangles are the nonconforming Morley finite element, the discontinuous Galerkin, the C0    interior penalty, and the WOPSIP schemes. Those methods are modified in their right-hand side and then are quasi-optimal in their respective discrete norms. The smoother JI M  is defined for a piecewise smooth input function by a (generalized) Morley interpolation I M  followed by a companion operator J. An abstract framework for the error analysis in the energy, weaker and piecewise Sobolev norms for the schemes is outlined and applied to the biharmonic equation. Three errors are also equivalent in some particular discrete norm from [Carstensen, Gallistl, Nataraj: Comparison results of nonstandard P 2  finite element methods for the biharmonic problem, ESAIM Math. Model. Numer. Anal. (2015)] without data oscillations. This paper extends and unifies the work [Veeser, Zanotti: Quasioptimal nonconforming methods for symmetric elliptic problems, SIAM J. Numer. Anal. 56 (2018)] to the discontinuous Galerkin scheme and adds error estimates in weaker and piecewise Sobolev norms.

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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