Lutz Angermann, Christian Henke
The paper presents results on piecewise polynomial approximations of tensor product type in Sobolev-Slobodecki spaces by various interpolation and projection techniques, on error estimates for quadrature rules and projection operators based on hierarchical bases, and on inverse inequalities. The main focus is directed to applications to discrete conservation laws.
Lutz Angermann, Christian Henke
We prove the $L^\infty(L^\infty)$-boundedness of a higher-order shock-capturing streamline-diffusion DG-method based on polynomials of degree $p\geq 0$ for general scalar conservation laws. The estimate is given for the case of several space dimensions and for conservation laws with initial and boundary conditions.
Lutz Angermann
A finite element approach for approximating the solution of a mathematical model for the response of a penetrable, bounded object (obstacle) to the excitation by an external electromagnetic field is presented and investigated. The model consists of a nonlinear Helmholtz equation that is reduced to a spherical domain. As a specific example, we consider a finite element method consisting of Courant-type elements with curved edges at the boundary of a circular computational domain in the two-dimensional case. We examine this method and more general conforming methods -- including three-dimensional ones -- with comparable properties for their well-posedness; in particular, the validity of a discrete inf-sup condition of the modified sesquilinear form uniformly with respect to both the truncation and the mesh parameters is shown. Under suitable assumptions to the nonlinearities, a quasi-optimal error estimate is obtained. Finally, the satisfiability of the approximation property of the finite element space required for the solvability of a class of adjoint linear problems is discussed.
Lutz Angermann
This article describes the extension of recent methods for a posteriori error estimation such as dual-weighted residual methods to node-centered finite volume discretizations of second order elliptic boundary value problems including upwind discretizations. It is shown how different sources of errors, in particular modeling errors and discretization errors, can be estimated with respect to a user-defined output functional.