Algebraic convergence for anisotropic edge elements in polyhedral domains

Abstract

We study approximation errors for the h-version of Nédélec edge elements on anisotropically refined meshes in polyhedra. Both tetrahedral and hexahedral elements are considered, and the emphasis is on obtaining optimal convergence rates in the H(curl) norm for higher order elements. Two types of estimates are presented: First, interpolation error estimates for functions in anisotropic weighted Sobolev spaces. Here we consider not only the H(curl)-conforming Nédélec elements, but also the H(div)-conforming Raviart-Thomas elements which appear naturally in the discrete version of the de Rham complex. Our technique is to transport error estimates from the reference element to the physical element via highly anisotropic coordinate transformations. Second, Galerkin error estimates for the standard H(curl) approximation of time harmonic Maxwell equations. Here we use the anisotropic weighted Sobolev regularity of the solution on domains with three-dimensional edges and corners. We also prove the discrete compactness property needed for the convergence of the Maxwell eigenvalue problem. Our results generalize those of [40] to the case of polyhedral corners and higher order elements.

The discrete compactness property for anisotropic edge elements on polyhedral domains

“…Our analysis has similarities with the ones developed in [4,16]. In particular, the key ingredients of our approach are:…”

“…The meshes considered in [4] satisfy our requirements. Therefore, we are extending the results of [4] to the case of tetrahedral meshes and elements of any order.…”

“…Therefore, we are extending the results of [4] to the case of tetrahedral meshes and elements of any order.…”

“…However, the approach allows to replace Lagrange by other scalar piecewise polynomials interpolations, changing eventually the mesh restrictions. This issue, which is a difference with the theory of [4], may be further analyzed in the future. In [4] the scalar interpolation that commutes with the edge interpolation in the De Rham diagram needed to be used.…”

“…However, the validity of the discrete compactness property on meshes adapted to edge or corner singularities was also considered. In [4] the discrete compactness property was obtained for edge elements on suitably refined meshes. This property combined with interpolation results was used to prove optimal algebraic convergence for both the source and eigenvalue problems.…”