Discrete Differential Forms, Approximation of Eigenvalue Problems, and Application to the p Version of Edge Finite Elements

Abstract

We are interested in the approximation of the eigenvalues of Hodge–Laplace operator in the framework of de Rham complex by using exterior calculus and suitable equivalent formulations in mixed form. We discuss the role of discrete compactness property and show how it is related to the classic conditions for the convergence of eigenvalues in mixed form. In this context, we review a recent result concerning the discrete compactness for the p version of discrete differential forms. One of the applications of the presented theory is the convergence analysis of the p version of edge finite elements for the approximation of Maxwell’s eigenvalues.