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We present, analyze, and test locally stabilized space–time finite element methods on fully unstructured simplicial space–time meshes for the numerical solution of space–time tracking parabolic optimal control problems with the standard L 2-regularization.We derive a priori discretization error estimates in terms of the local mesh-sizes for shape-regular meshes. The adaptive version is driven by local residual error indicators, or, alternatively, by local error indicators derived from a new functional a posteriori error estimator. The latter provides a guaranteed upper bound of the error, but is more costly than the residual error indicators. We perform numerical tests for benchmark examples having different features. In particular, we consider a discontinuous target in form of a first expanding and then contracting ball in 3d that is fixed in the 4d space– time cylinder.

We consider locally stabilized, conforming finite element schemes on completely unstructured simplicial space-time meshes for the numerical solution of parabolic initial-boundary value problems with variable, possibly discontinuous in space and time, coefficients. Distributional sources are also admitted. Discontinuous coefficients, non-smooth boundaries, changing boundary conditions, non-smooth or incompatible initial conditions, and non-smooth right-hand sides can lead to non-smooth solutions. We present new a priori and a posteriori error estimates for low-regularity solutions. In order to avoid reduced convergence rates appearing in the case of uniform mesh refinement, we also consider adaptive refinement procedures based on residual a posteriori error indicators and functional a posteriori error estimators. The huge system of space-time finite element equations is then solved by means of GMRES preconditioned by space-time algebraic multigrid. In particular, in the 4d space-time case that is 3d in space, simultaneous space-time parallelization can considerably reduce the computational time. We present and discuss numerical results for several examples possessing different regularity features., 23 pages, 11 figures

Erworben im Rahmen der Schweizer Nationallizenzen (http://www.nationallizenzen.ch), The paper is related to a coupled problem describing piezoelectric effects in an elastic body. For this problem, we deduce majorants of the distance between the exact solution and any approximation in the respective energy class of functions satisfying the boundary conditions. The majorants are fully computable and do not contain mesh dependent constants. They vanish if and only if an approximate solution coincides with the exact one and provide guaranteed upper bounds of errors in terms of the natural energy norm associated with the coupled problem studied.

This work is devoted to fast and parameter-robust iterative solvers for frequency domain finite element equations, approximating the time-periodic eddy current problem with multiharmonic or time-periodic excitations in time. We construct a preconditioned MinRes solver for the frequency domain equations, that is robust with respect to the discretization parameters as well as all involved “bad” parameters like the conductivity, the reluctivity and possible regularization parameters.

This publication is partially based on work supported by Award No. KUS-C1-016-04, made by King Abdullah University of Science and Technology (KAUST), and by the Austrian Science Fund 'Fonds zur Forderung der wissenschaftlichen Forschung (FWF)' under the grant P19255.