On high-order Nedelec elements and Maxwell Eigensolvers using inexact projection

Sabine Zaglmayr and J. Schoeberl

Email: sz@jku.at
Postal address: Radoninstitut (RICAM), Altenbergerstrasse 69, 4040 Linz, Austria

The goal of the presented work is the efficient computation of Maxwell eigenvalue problems using high-order H(curl)-conforming finite elements.

In the first part, we present a general, unified construction principle for H(curl)-conforming finite elements of variable and arbitrary order. In order to allow for geometric $h$-refinement, we consider hybrid meshes, involving hexahedral, tetrahedral, and prismatic elements.  The keypoint of our framework is to respect the exact de Rham sequence already in the construction of the FE basis functions. 

A short outline of the construction is as follows. We start with the classical lowest-order Nedelec shape functions. Then we take the gradients of edge-based, face-based and cell-based shape functions of the higher-order $H^1$-conforming FE-space. Finally, we hierarchically extend these sets of functions to a conforming basis of the desired polynomial space. 

By our separate treatment of the edge-based, face-based, and cell-based functions, and by including the corresponding gradient functions, we can establish the local exact sequence property: the subspaces corresponding to a single edge, a single face or a single cell already form an exact sequence. A main advantage is that we can choose an arbitrary polynomial order on each edge, face, and cell independently, without destroying the global exact sequence property.  

Further on, simple block ASM-preconditioning gets efficient and the implementation of the discrete gradient-operator from $H^1$ to H(curl) requires no additional computational costs.

In the second part, we focus on computing the few lowest eigen-pairs of the Maxwell eigenvalue problem.  We use the subspace version of the locally optimal preconditioned conjugate gradient method (LOBPCG), which requires a good preconditioner for the curl-curl problem. Since the desired eigenfunctions belong to the orthogonal complement of the gradient functions,  we have to extend the iterative scheme by an orthogonal projection in each iteration step. This requires the solution of a potential problem, which can be done approximately by a couple of PCG-iterations. Considering benchmark problems involving highly singular eigensolutions, we demonstrate the performance of the constructed preconditioners and the eigenvalue solver in combination with hp-discretization on geometrically refined, anisotropic meshes.