A SKELETAL FINITE ELEMENT METHOD CAN COMPUTE LOWER EIGENVALUE BOUNDS

The skeletal finite element method (FEM) in this paper is a hybridized discontinuous Galerkin FEM with the Lehrenfeld-Schiiberl stabilization and also known as a weak Galerkin FEM. With an appropriate stabilization, it provides eigenvalue approximations for the Laplacian on any regular triangulation T with maximal mesh-size h(max), which are guaranteed lower eigenvalue bounds (GLB) if they are sufficiently large. This paper establishes a bound alpha(T) for a global stabilization parameter alpha such that alpha <= alpha(T) leads to an eigenvalue approximation lambda(h) <= lambda for the exact eigenvalue lambda, provided kappa(2)(CR)h(max)(2) lambda(h) <= 1 for a universal constant kappa(CR). For a 2D triangulation T into triangles, a comparison with the bound CRGLB := lambda(CR)/(1 + epsilon lambda(CR)) <= lambda from [C. Carstensen and J. Gedicke, Math. Comp., 83 (2014), pp. 2605-2629] proves under the same conditions that CRGLB <= lambda(h) <= lambda. The paper also provides an alternative proof of the already established asymptotic lower bound property.