Adaptive $hp$-FEM with Arbitrary-Level Hanging Nodes for Maxwell's Equations

Abstract

Adaptive higher-order finite element methods ($hp$-FEM) are well known for their potential of exceptionally fast (exponential) convergence. However, most $hp$-FEM codes remain in an academic setting due to an extreme algorithmic complexity of $hp$-adaptivity algorithms. This paper aims at simplifying $hp$-adaptivity for $H$(curl)-conforming approximations by presenting a novel technique of arbitrary-level hanging nodes. The technique is described and it is demonstrated numerically that it makes adaptive $hp$-FEM more efficient compared to $hp$-FEM on regular meshes and meshes with one-level hanging nodes.