Volume 18, Issue 4

We consider stochastic elliptic variational inequalities of the second kind involving a bilinear form with stochastic diffusion coefficient. We prove existence and uniqueness of weak solutions, propose a stochastic Galerkin approximation of an equivalent parametric reformulation, and show equivalence to a related collocation method. Numerical experiments illustrate the efficiency of our approach and suggest similar error estimates as for linear elliptic problems.

The convolution , where ƒ is smooth, except for some local singularities, arises for example in electronic structure calculations. An efficient convolution with the Newton potential in d dimensions has been proposed in [Hackbusch, Numer. Math. 110: 449–489, 2008]. The convolution is approximated on a refined grid and additional approximations are introduced for efficient evaluation. This paper studies the performance of the method and a precise error analysis of the method is discussed.

We consider an adaptive finite element method (AFEM) for the Laplace eigenvalue problem in bounded polygonal or polyhedral domains. We provide an a posteriori error analysis based on a residual type estimator which consists of element and face residuals. The a posteriori error analysis further involves an oscillation term. We prove a reduction in the energy norm of the discretization error and the oscillation term. Numerical results are given illustrating the performance of the AFEM.

This paper develops a combined a posteriori analysis for the discretization and iteration errors in the solution of elliptic eigenvalue problems by the finite element method. The emphasis is on the iterative solution of the discretized eigenvalue problem by a Krylov-space method. The underlying theoretical framework is that of the Dual Weighted Residual (DWR) method for goal-oriented error estimation. On the basis of computable a posteriori error estimates the algebraic iteration can be adjusted to the discretization within a successive mesh adaptation process. The functionality of the proposed method is demonstrated by numerical examples.