Space-Time Finite Element Methods for Parabolic Initial-Boundary Value Problems with Non-smooth Solutions

Andreas Schafelner

June 4, 2019, 4:30 p.m. S2 416-1

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, and changing boundary conditions, non-smooth initial conditions, and non-smooth right-hand sides can lead to non-smooth solutions. For instance, in electromagnetics, permanent magnets cause line-delta-distributions in the source term of the right-hand side in 2D quasi-magnetostatic simulations of electrical machines. We present new a priori estimates under the assumption of local maximal parabolic regularity that includes low-regularity solutions arising from non-smooth data such as mentioned above. 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, that is positive definite, but non-symmetric, is then solved by means of the Generalized Minimal Residual Method 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 with the library MFEM. We present numerical examples with different features. The numerical results nicely confirm our theoretical findings. The parallel version of the code shows an excellent parallel performance.