Abstract:
We present various results on the equivalence and mapping properties under affine transformations of fractional-order Sobolev norms and semi-norms of orders between zero and one. Main results are mutual estimates of the three semi-norms of Sobolev-Slobodeckij, interpolation and quotient space types. In particular, we show that the former two are uniformly equivalent under affine mappings that ensure shape regularity of the domains under consideration.

Abstract:
We establish a mortar boundary element scheme for hypersingular boundary integral equations representing elliptic boundary value problems in three dimensions. We prove almost quasi-optimal convergence of the scheme in broken Sobolev norms of order 1/2. Sub-domain decompositions can be geometrically non-conforming and meshes must be quasi-uniform only on sub-domains. Numerical results confirm the theory.

Abstract:
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.

Abstract:
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.

Abstract:
For the non-conforming Crouzeix-Raviart boundary elements from [Heuer, Sayas: Crouzeix-Raviart boundary elements, Numer. Math. 112, 2009], we develop and analyze a posteriori error estimators based on the $h-h/2$ methodology. We discuss the optimal rate of convergence for uniform mesh refinement, and present a numerical experiment with singular data where our adaptive algorithm recovers the optimal rate while uniform mesh refinement is sub-optimal. We also discuss the case of reduced regularity by standard geometric singularities to conjecture that, in this situation, non-uniformly refined meshes are not superior to quasi-uniform meshes for Crouzeix-Raviart boundary elements.

Abstract:
We present and analyze a discontinuous Petrov-Galerkin method with optimal test functions for a reaction-dominated diffusion problem in two and three space dimensions. We start with an ultra-weak formulation that comprises parameters $\alpha$, $\beta$ to allow for general $\varepsilon$-dependent weightings of three field variables ($\varepsilon$ being the small diffusion parameter). Specific values of $\alpha$ and $\beta$ imply robustness of the method, that is, a quasi-optimal error estimate with a constant that is independent of $\varepsilon$. Moreover, these values lead to a norm for the field variables that is known to be balanced in $\varepsilon$ for model problems with typical boundary layers. Several numerical examples underline our theoretical estimates and reveal stability of approximations even for very small $\varepsilon$.

Abstract:
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 H(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 H^{-1/2}(div)-conforming p-interpolation operator that assumes only H^r \cap H^{-1/2}(div)-regularity (r > 0) and for which we show quasi-stability with respect to polynomial degrees.

Abstract:
We present a simple discretization by radial basis functions for the Poisson equation with Dirichlet boundary condition. A Lagrangian multiplier using piecewise polynomials is used to accommodate the boundary condition. This simplifies previous attempts to use radial basis functions in the interior domain to approximate the solution and on the boundary to approximate the multiplier, which technically requires that the mesh norm in the interior domain is significantly smaller than that on the boundary. Numerical experiments confirm theoretical results.

Abstract:
We present an ultra-weak formulation of a hypersingular integral equation on closed polygons and prove its well-posedness and equivalence with the standard variational formulation. Based on this ultra-weak formulation we present a discontinuous Petrov-Galerkin method with optimal test functions and prove its quasi-optimal convergence in $L^2$. Theoretical results are confirmed by numerical experiments on an open curve with uniform and adaptively refined meshes.

Abstract:
Generalizing the framework of an ultra-weak formulation for a hypersingular integral equation on closed polygons in [N. Heuer, F. Pinochet, arXiv 1309.1697 (to appear in SIAM J. Numer. Anal.)], we study the case of a hypersingular integral equation on open and closed polyhedral surfaces. We develop a general ultra-weak setting in fractional-order Sobolev spaces and prove its well-posedness and equivalence with the traditional formulation. Based on the ultra-weak formulation, we establish a discontinuous Petrov-Galerkin method with optimal test functions and prove its quasi-optimal convergence in related Sobolev norms. For closed surfaces, this general result implies quasi-optimal convergence in the L^2-norm. Some numerical experiments confirm expected convergence rates.