Adaptive Space-Time Finite Element and Isogeometric Analysis

The traditional approaches to the numerical solution of initial‚boundary value prob‚ lems ̆IBVP ̄ for parabolic or hyperbolic Partial Differential Equations ̆PDEs ̄ are based on the separation of the discretization in time and space leading to time‚ stepping methods; see, e.g., [20]. This separation of time and space discretizations comes along with some disadvantages with respect to parallelization and adaptiv‚ ity. To overcome these disadvantages, we consider completely unstructured finite element ̆fe ̄ or isogeometric ̆B‚spline or NURBS ̄ discretizations of the space‚ time cylinder and the corresponding stable space‚time variational formulations of the IBVP under consideration. Unstructured space‚time discretizations considerably facilitate the parallelization and the simultaneous space‚time adaptivity. Moving spatial domains or interfaces can easily be treated since they are fixed in the space‚ time cylinder. Beside initial‚boundary value problems for parabolic PDEs, we will also consider optimal control problems constrained by linear or non‚linear parabolic PDEs. Here unstructured space‚time methods are especially suited since the reduced optimality system couples two parabolic equations for the state and adjoint state that are forward and backward in time, respectively. In contrast to time‚stepping methods, one has to solve one big linear or non‚linear system of algebraic equations. Thus, the memory requirement is an issue. In this connection, adaptivity, parallelization, and matrix‚free implementations are very important techniques to overcome this bottleneck. Fast parallel solvers like domain decomposition and multigrid solvers are the most important ingredients of efficient space‚time methods. This paper is partially based on joint works with Svetlana Kyas ̆Matculevich ̄ and Sergey Repin on adaptive space‚time IGA based on functional a posteriori error es‚ timators [10, 11], Martin Neumüller and Andreas Schafelner on adaptive space‚time