1. A SKELETAL FINITE ELEMENT METHOD CAN COMPUTE LOWER EIGENVALUE BOUNDS.
- Author
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CARSTENSEN, CARSTEN, QILONG ZHAI, and RAN ZHANG
- Subjects
- *
FINITE element method , *TRIANGULATION - Abstract
The skeletal finite element method (FEM) in this paper is a hybridized discontinuous Galerkin FEM with the Lehrenfeld--Sch\"oberl stabilization and also known as a weak Galerkin FEM. With an appropriate stabilization, it provides eigenvalue approximations for the Laplacian on any regular triangulation \scrT with maximal mesh-size hmax, which are guaranteed lower eigenvalue bounds (GLB) if they are sufficiently large. This paper establishes a bound \alpha (\scrT) for a global stabilization parameter \alpha such that \alpha \leq \alpha (\scrT) leads to an eigenvalue approximation \lambda h \leq \lambda for the exact eigenvalue \lambda, provided \kappa 2 CRh2 max \lambda h \leq 1 for a universal constant \kappa CR. For a 2D triangulation \scrT into triangles, a comparison with the bound CRGLB := \lambda CR/(1 + \varepsilon \lambda CR) \leq \lambda from [C. Carstensen and J. Gedicke, Math. Comp., 83 (2014), pp. 2605--2629] proves under the same conditions that CRGLB \leq \lambda h \leq \lambda. The paper also provides an alternative proof of the already established asymptotic lower bound property. [ABSTRACT FROM AUTHOR]
- Published
- 2020
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