Discrete compactness and the approximation of Maxwell's equations in $\mathbb{R}^3$

Abstract

Abstract. We analyze the use of edge finite element methods to approximate Maxwell's equations in a bounded cavity. Using the theory of collectively compact operators, we prove h-convergence for the source and eigenvalue problems. This is the first proof of convergence of the eigenvalue problem for general edge elements, and it extends and unifies the theory for both problems. The convergence results are based on the discrete compactness property of edge element due to Kikuchi. We extend the original work of Kikuchi by proving that edge elements of all orders possess this property.

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“…We make the assumption that subdomains Ω j are convex only so that we can invoke Proposition 5.1 in Chapter III of [13] to obtain the Poincaré inequality (2.4) for discretely divergence free functions on the (convex) reference domains Ω i (see (4.3) in particular). Recent results indicate that such an inequality holds under weaker assumptions [1,19]. Our analysis remains unchanged if these assumptions are made instead of convexity.…”

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“…In spherical coordinates, the matrix function µ is represented by multiplication by a diagonal matrix with complex diagonal entries {D 11 …”

“…Our analysis uses many of the ideas developed for curl-conforming finite element approximations to time-harmonic electromagnetic problems on bounded domains [7,9,10,11]. We use a duality approach reminiscent of the argument of Schatz [14] with modifications to handle the lack of uniform ellipticity similar to those used in [6,9].…”

Discrete compactness for a discontinuous Galerkin approximation of Maxwell's system

“…This property has been studied for standard Galerkin approximation for a quite large family of edge elements on two and three dimensional domains [7,8,15,21,26]. But to our knowledge this property is not yet proved for the discontinuous Galerkin method.…”

“…Our proof of the discrete compactness property is based on the same property for the standard Galerkin approximation proved in [26] and the use of a decomposition of the discontinuous approximation space into a continuous one and its orthogonal for an appropriate inner product similar to [19,20] (but different from the one used in these references). The discrete Friedrichs inequality follows from this discrete compactness property and a contradiction argument.…”