Optimal error estimation for H(curl)-conforming p-interpolation in two dimensions

In this paper we prove an optimal error estimate for the H(curl)-conforming projection based p-interpolation operator introduced in [L. Demkowicz and I. Babuska, p interpolation error estimates for edge finite elements of variable order in two dimensions, SIAM J. Numer. Anal., 41 (2003), pp. 1195-1208]. This result is proved on the reference element (either triangle or square) K for regular vector fields in H^r(curl,K) with arbitrary r>0. The formulation of the result in the H(div)-conforming setting, which is relevant for the analysis of high-order boundary element approximations for Maxwell's equations, is provided as well

Natural hp-BEM for the electric field integral equation with singular solutions

We apply the hp-version of the boundary element method (BEM) for the numerical solution of the electric field integral equation (EFIE) on a Lipschitz polyhedral surface G. The underlying meshes are supposed to be quasi-uniform triangulations of G, and the approximations are based on either Raviart-Thomas or Brezzi-Douglas-Marini families of surface elements. Non-smoothness of G leads to singularities in the solution of the EFIE, severely affecting convergence rates of the BEM. However, the singular behaviour of the solution can be explicitly specified using a finite set of power functions (vertex-, edge-, and vertex-edge singularities). In this paper we use this fact to perform an a priori error analysis of the hp-BEM on quasi-uniform meshes. We prove precise error estimates in terms of the polynomial degree p, the mesh size h, and the singularity exponents.Comment: 17 page

The hp-BEM with quasi-uniform meshes for the electric field integral equation on polyhedral surfaces: a priori error analysis

This paper presents an a priori error analysis of the hp-version of the boundary element method for the electric field integral equation on a piecewise plane (open or closed) Lipschitz surface. We use H(div)-conforming discretisations with Raviart-Thomas elements on a sequence of quasi-uniform meshes of triangles and/or parallelograms. Assuming the regularity of the solution to the electric field integral equation in terms of Sobolev spaces of tangential vector fields, we prove an a priori error estimate of the method in the energy norm. This estimate proves the expected rate of convergence with respect to the mesh parameter h and the polynomial degree p

On the convergence of the hp-BEM with quasi-uniform meshes for the electric field integral equation on polyhedral surfaces

In this paper the hp-version of the boundary element method is applied to the electric field integral equation on a piecewise plane (open or closed) Lipschitz surface. The underlying meshes are supposed to be quasi-uniform. We use \bH(\div)-conforming discretisations with quadrilateral elements of Raviart-Thomas type and establish quasi-optimal convergence of hp-approximations. Main ingredient of our analysis is a new \tilde\bH^{-1/2}(\div)-conforming p-interpolation operator that assumes only \bH^r\cap\tilde\bH^{-1/2}(\div)-regularity ($r>0$) and for which we show quasi-stability with respect to polynomial degrees

Generalized Finite Element Systems for smooth differential forms and Stokes problem

We provide both a general framework for discretizing de Rham sequences of differential forms of high regularity, and some examples of finite element spaces that fit in the framework. The general framework is an extension of the previously introduced notion of Finite Element Systems, and the examples include conforming mixed finite elements for Stokes' equation. In dimension 2 we detail four low order finite element complexes and one infinite family of highorder finite element complexes. In dimension 3 we define one low order complex, which may be branched into Whitney forms at a chosen index. Stokes pairs with continuous or discontinuous pressure are provided in arbitrary dimension. The finite element spaces all consist of composite polynomials. The framework guarantees some nice properties of the spaces, in particular the existence of commuting interpolators. It also shows that some of the examples are minimal spaces.Comment: v1: 27 pages. v2: 34 pages. Numerous details added. v3: 44 pages. 8 figures and several comments adde