Monika Wolfmayr

Ulrich Langer, Sergey Repin, Monika Wolfmayr

This paper is devoted to the a posteriori error analysis of multiharmonic finite element approximations to distributed optimal control problems with time-periodic state equations of parabolic type. We derive a posteriori estimates of functional type, which are easily computable and provide guaranteed upper bounds for the state and co-state errors as well as for the cost functional. These theoretical results are confirmed by several numerical tests that show high efficiency of the a posteriori error bounds.

Jari Toivanen, Monika Wolfmayr

This paper is devoted to the efficient numerical solution of the Helmholtz equation in a two- or three-dimensional rectangular domain with an absorbing boundary condition (ABC). The Helmholtz problem is discretized by standard bilinear and trilinear finite elements on an orthogonal mesh yielding a separable system of linear equations. The main key to high performance is to employ the Fast Fourier transform (FFT) within a fast direct solver to solve the large separable systems. The computational complexity of the proposed FFT based direct solver is O(N log N) operations. Numerical results for both two- and three-dimensional problems are presented confirming the efficiency of the method discussed.

Svetlana Matculevich, Monika Wolfmayr

This work is aimed at the derivation of reliable and efficient a posteriori error estimates for convection-dominated diffusion problems motivated by a linear Fokker-Planck problem appearing in computational neuroscience. We obtain computable error bounds of the functional type for the static and time-dependent case and for different boundary conditions (mixed and pure Neumann boundary conditions). Finally, we present a set of various numerical examples including discussions on mesh adaptivity and space-time discretisation. The numerical results confirm the reliability and efficiency of the error estimates derived.

Monika Wolfmayr

In this work, new theoretical results on functional type a posteriori estimates for elliptic optimal control problems with control constraints are presented. More precisely, we derive new, sharp, guaranteed and fully computable lower bounds for the cost functional in addition to the already existing upper bounds. Using both, the lower and the upper bounds, we arrive at two-sided estimates for the cost functional. We prove that these bounds finally lead to sharp, guaranteed and fully computable upper estimates for the discretization error in the state and the control of the optimal control problem. First numerical tests are presented confirming the efficiency of the a posteriori estimates derived.

Ulrich Langer, Sergey Repin, Monika Wolfmayr

The paper is concerned with parabolic time-periodic boundary value problems which are of theoretical interest and arise in different practical applications. The multiharmonic finite element method is well adapted to this class of parabolic problems. We study properties of multiharmonic approximations and derive guaranteed and fully computable bounds of approximation errors. For this purpose, we use the functional a posteriori error estimation techniques earlier introduced by S. Repin. Numerical tests confirm the efficiency of the a posteriori error bounds derived.