Volume 13, Issue 1

Abstract. This work is devoted to fast and parameter-robust iterative solvers for frequency domain finite element equations, approximating the time-periodic eddy current problem with multiharmonic or time-periodic excitations in time. We construct a preconditioned MinRes solver for the frequency domain equations, that is robust with respect to the discretization parameters as well as all involved “bad” parameters like the conductivity, the reluctivity and possible regularization parameters.

Abstract. We consider the finite element approximation of the solution to a singularly perturbed second order differential equation with a constant delay. The boundary value problem can be cast as a singularly perturbed transmission problem, whose solution may be decomposed into a smooth part, a boundary layer part, an interior/interface layer part and a remainder. Upon discussing the regularity of each component, we show that under the assumption of analytic input data, the hp version of the finite element method on an appropriately designed mesh yields robust exponential convergence rates. Numerical results illustrating the theory are also included.

Abstract. In the present paper it is shown that the interpolation problem for multiple knot cardinal splines subject to general interpolation conditions has a unique solution with polynomial growth if the data grow correspondingly provided a certain determinantal condition is satisfied. An application to H s error estimates for the interpolation with periodic multiple knot splines is given.

Abstract. The Galerkin and SDFEM methods are compared for a steady state convection problem. The theoretical part of this work deals with the development of approximation results for continuous solutions on the unit square containing an edge singularity. In the numerical part we verify those approximation results by considering continuous as well as discontinuous solutions to the transport problem on an annular domain with a singularity at the inner circle.

Abstract. The local discontinuous Galerkin method (LDG) is considered for solving one-dimensional singularly perturbed two-point boundary value problems of reaction-diffusion type. Pointwise error estimates for the LDG approximation to the solution and its derivative are established on a Shishkin-type mesh. Numerical experiments are presented. Moreover, a superconvergence of order of the numerical traces is observed numerically.

Abstract. This paper presents the first feasible method for the approximation of solution sets of semi-linear elliptic partial differential inclusions. It is based on a new Galerkin Finite Element approach that projects the original differential inclusion to a finite-dimensional subspace of . The problem that remains is to discretize the unknown solution set of the resulting finite-dimensional algebraic inclusion in such a way that efficient algorithms for its computation can be designed and error estimates can be proved. One such discretization and the corresponding basic algorithm are presented along with several enhancements, and the algorithm is applied to two model problems.