Discrete Compactness for p-Version of Tetrahedral Edge Elements

TL;DR

This paper proves a long-standing conjecture that certain finite element spaces for Maxwell problems maintain a key compactness property as the polynomial degree increases, ensuring accurate spectral approximations.

Contribution

It establishes the discrete compactness property for the p-version of tetrahedral edge elements, confirming spectral correctness for Maxwell eigenvalue approximations.

Findings

Discrete compactness holds as p→∞ for these elements.

Spectral Galerkin methods are asymptotically correct.

Supports reliable high-order finite element analysis for electromagnetics.

Abstract

We consider the first family of $\Hcurl$-conforming Ned\'el\'ec finite elements on tetrahedral meshes. Spectral approximation ($p$-version) is achieved by keeping the mesh fixed and raising the polynomial degree $p$ uniformly in all mesh cells. We prove that the associated subspaces of discretely weakly divergence free piecewise polynomial vector fields enjoy a long conjectured discrete compactness property as $p\to\infty$. This permits us to conclude asymptotic spectral correctness of spectral Galerkin finite element approximations of Maxwell eigenvalue problems.