Volume 22, Issue 1

It is well-known that the performance of snapshot based POD and POD-DEIM for spatially semidiscretized parabolic PDEs depends on the proper selection of the snapshot locations. In this contribution, we present an approach that for a fixed number of snapshots selects the location based on error equilibration in the sense that the global discretization error is approximately the same in each associated subinterval. The global discretization error is assessed by a hierarchical-type a posteriori error estimator known from automatic time-stepping for systems of ODEs. We study the impact of this snapshot selection on error equilibration for the ROM and provide numerical examples that illustrate the performance of the suggested approach.

In a previous part I, we introduced a method for the evaluation of singular integrals arising in the discretization of integral equations based on the repeated subdivision of domains. The integrals defined on these subdomains are classified such that a class of integrals can be expressed as a sum of regular integrals and representatives of other classes. A system of equations describes the relations between the classes. Furthermore the approximate value of the singular integrals only depends on the accuracy of the calculation of regular integrals. Part I left a gap on certain parameter configurations on which the mentioned system of equations is irregular. This paper shall close this gap. To this end, we introduce an alternative splitting strategy based on a modified Hadamard partie finie integral.

In this paper we present a survey of numericalmethods for optimal control problems governed by convection-diffusion equations. For ease of exposition we focus on stationary convection-diffusion optimal control problems with distributed control. Several effective numericalmethods including stabilizedmethods and discontinuous Galerkinmethods are reviewed. Furthermore, we also make some investigations about convection-diffusion optimal control problems with state constraints and pointwisely imposed control.