2020/03/20 by Ulrich Langer, Langer, Ulrich, Andreas Schafelner +1
Engineering · Mathematics · #35K20 #65M15 #65M50 #65M60 #65Y05 #Advanced Numerical Methods in Computational Mathematics #Computational Fluid Dynamics and Aerodynamics #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods for differential equations
paper · pdf · doi:10.48550/arxiv.2003.09248
openalex publication_date 2020/03/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider locally stabilized, conforming finite element schemes on\ncompletely unstructured simplicial space-time meshes for the numerical solution\nof parabolic initial-boundary value problems with variable, possibly\ndiscontinuous in space and time, coefficients. Distributional sources are also\nadmitted. Discontinuous coefficients, non-smooth boundaries, changing boundary\nconditions, non-smooth or incompatible initial conditions, and non-smooth\nright-hand sides can lead to non-smooth solutions. We present new a priori and\na posteriori error estimates for low-regularity solutions. In order to avoid\nreduced convergence rates appearing in the case of uniform mesh refinement, we\nalso consider adaptive refinement procedures based on residual a posteriori\nerror indicators and functional a posteriori error estimators. The huge system\nof space-time finite element equations is then solved by means of GMRES\npreconditioned by space-time algebraic multigrid. In particular, in the 4d\nspace-time case that is 3d in space, simultaneous space-time parallelization\ncan considerably reduce the computational time. We present and discuss\nnumerical results for several examples possessing different regularity\nfeatures.\n