Alexei Bespalov

The p -version of the boundary element method for weakly singular operators on piecewise plane open surfaces

We study piecewise polynomial approximations in negative order Sobolev norms of singularities which are inherent to Neumann data of elliptic problems of second order in polyhedral domains. The worst case of exterior crack problems in three dimensions is included. As an application, we prove an optimal a priori error estimate for the p-version of the BEM with weakly singular operators on polyhedral Lipschitz surfaces and piecewise plane open screens.

On the finite element method for elliptic problems with degenerate and singular coefficients

We consider Dirichlet boundary value problems for second order elliptic equations over polygonal domains. The coefficients of the equations under consideration degenerate at an inner point of the domain, or behave singularly in the neighborhood of that point. This behavior may cause singularities in the solution. The solvability of the problems is proved in weighted Sobolev spaces, and their approximation by finite elements is studied. This study includes regularity results, graded meshes, and inverse estimates. Applications of the theory to some problems appearing in quantum mechanics are given. Numerical results are provided which illustrate the theory and confirm the predicted rates of convergence of the finite element approximations for quasi-uniform meshes.

Energy Norm A Posteriori Error Estimation for Parametric Operator Equations

Stochastic Galerkin approximation is an increasingly popular approach for the solution of elliptic PDE problems with correlated random data. A typical strategy is to combine conventional (

Convergence of the Natural

h

hp

-)finite element approximation on the spatial domain with spectral (

-BEM for the Electric Field Integral Equation on Polyhedral Surfaces

p

A priori error analysis of stochastic galerkin mixed approximations of elliptic pdes with random data

-)approximation on a finite-dimensional manifold in the (stochastic) parameter domain. The issues involved in a posteriori error analysis of computed solutions are outlined in this paper using an abstract setting of parametric operator equations. A novel energy error estimator that uses a parameter-free part of the underlying differential operator is introduced which effectively exploits the tensor product structure of the approximation space. We prove that our error estimator is reliable and efficient. We also discuss different strategies for enriching the approximation space and prove two-sided estimates of the error reduction for the corresponding enhanced approximations. These give computable estimates of the error reduction that depend only on...

Optimal Error Estimation for

We consider the variational formulation of the electric field integral equation on bounded polyhedral open or closed surfaces. We employ a conforming Galerkin discretization based on

{\bfH}({\rmcurl})

\mathrm{div}_{\Gamma}

-Conforming

-conforming Raviart-Thomas boundary elements of locally variable polynomial degree on shape-regular surface meshes. We establish asymptotic quasi optimality of Galerkin solutions on sufficiently fine meshes or for sufficiently high polynomial degrees.