Space-Time Finite Element Solvers of Parabolic Evolution Problems with Non-smooth Solutions

Andreas Schafelner

Jan. 29, 2019, 3:30 p.m. S2 054

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 coefficients that may be discontinuous in space and time. 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 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. The huge system of space-time finite element equations is then solved by means of GMRES preconditioned by algebraic multigrid. In particular, in the 4D space-time case that is 3D in space, simultaneous space-time adaptivity and parallelization can considerably reduce the computational time. The space-time finite element solver was implemented in the framework of MFEM. We present different numerical experiments. The numerical results nicely confirm our theoretical findings. The parallel version of the code shows an excellent parallel performance.