Dominik Hafemeyer

Dominik Hafemeyer, Christian Kahle, Johannes Pfefferer

This work is concerned with quasi-optimal a-priori finite element error estimates for the obstacle problem in the $L^2$-norm. The discrete approximations are introduced as solutions to a finite element discretization of an accordingly regularized problem. The underlying domain is only assumed to be convex and polygonally or polyhedrally bounded such that an application of point-wise error estimates results in a rate less than two in general. The main ingredient for proving the quasi-optimal estimates is the structural and commonly used assumption that the obstacle is inactive on the boundary of the domain. Then localization techniques are used to estimate the global $L^2$-error by a local error in the inner part of the domain, where higher regularity for the solution can be assumed, and a global error for the Ritz-projection of the solution, which can be estimated by standard techniques. We validate our results by numerical examples.

Sebastian Engel, Dominik Hafemeyer, Christian Münch et al.

In this work we discuss the problem of identifying sound sources from pressure measurements with a Bayesian approach. The acoustics are modelled by the Helmholtz equation and the goal is to get information about the number, strength and position of the sound sources, under the assumption that measurements of the acoustic pressure are noisy. We propose a problem specific prior distribution of the number, the amplitudes and positions of the sound sources and algorithms to compute an approximation of the associated posterior. We also discuss a finite element discretization of the Helmholtz equation for the practical computation and prove convergence rates of the resulting discretized posterior to the true posterior. The theoretical results are illustrated by numerical experiments, which indicate that the proven rates are sharp.