Volume 27, Issue 2

We consider the finite element approximation of fractional powers of regularly accretive operators via the Dunford–Taylor integral approach. We use a sinc quadrature scheme to approximate the Balakrishnan representation of the negative powers of the operator as well as its finite element approximation. We improve the exponentially convergent error estimates from [A. Bonito and J. E. Pasciak, IMA J. Numer. Anal ., 37 (2016), No. 3, 1245–1273] by reducing the regularity required on the data. Numerical experiments illustrating the new theory are provided.

In this paper, we present a general class of exponentially fitted two-step peer methods for the numerical integration of ordinary differential equations. The numerical scheme is constructed in order to exploit a-priori known information about the qualitative behaviour of the solution by adapting peer methods already known in literature. Examples of methods with 2 and 3 stages are provided. The effectiveness of this problem-oriented approach is shown through some numerical tests on well-known problems.

In the context of unfitted finite element discretizations the realization of high order methods is challenging due to the fact that the geometry approximation has to be sufficiently accurate. Recently a new unfitted finite element method was introduced which achieves a high order approximation of the geometry for domains which are implicitly described by smooth level set functions. This method is based on a parametric mapping which transforms a piecewise planar interface (or surface) reconstruction to a high order approximation. In the paper [C. Lehrenfeld and A. Reusken, IMA J. Numer. Anal . 38 (2018), No. 3, 1351–1387] an a priori error analysis of the method applied to an interface problem is presented. The analysis reveals optimal order discretization error bounds in the H 1 -norm. In this paper we extend this analysis and derive optimal L 2 -error bounds.

The paper presents a goal-oriented error control based on the dual weighted residual method (DWR) for the finite cell method (FCM), which is characterized by an enclosing domain covering the domain of the problem. The error identity derived by the DWR method allows for a combined treatment of the discretization and quadrature error introduced by the FCM. We present an adaptive strategy with the aim to balance these two error contributions. Its performance is demonstrated for several two-dimensional examples.