Asymptotic optimality of the edge finite element approximation of the time-harmonic Maxwell's equations

TL;DR

This paper proves that Nédélec edge finite element methods for time-harmonic Maxwell's equations are asymptotically optimal, providing bounds on approximation errors and stability that hold under minimal regularity assumptions.

Contribution

It establishes the asymptotic optimality and discrete inf-sup stability of edge finite element approximations for Maxwell's equations under minimal regularity conditions.

Findings

Error bounds approach the best approximation as mesh refines

Discrete inf-sup stability constant close to continuous constant

Results hold for general domains and material properties

Abstract

We analyze the conforming approximation of the time-harmonic Maxwell's equations using N\'ed\'elec (edge) finite elements. We prove that the approximation is asymptotically optimal, i.e., the approximation error in the energy norm is bounded by the best-approximation error times a constant that tends to one as the mesh is refined and/or the polynomial degree is increased. Moreover, under the same conditions on the mesh and/or the polynomial degree, we establish discrete inf-sup stability with a constant that corresponds to the continuous constant up to a factor of two at most. Our proofs apply under minimal regularity assumptions on the exact solution, so that general domains, material coefficients, and right-hand sides are allowed.