Abstract 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 are possibly discontinuous in space and time. 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 rates of convergence that appear when performing 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, simultaneous space-time parallelization can considerably reduce the computational time. We present and discuss numerical results for several examples possessing different regularity features.

中文翻译：

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具有分布源的非自治抛物线问题的自适应时空有限元方法

摘要 我们在完全非结构化的单纯时空网格上考虑局部稳定、一致的有限元方案，用于求解具有可能在空间和时间上不连续的可变系数的抛物线初边界值问题。分布式来源也被承认。不连续的系数、不光滑的边界、不断变化的边界条件、不光滑或不兼容的初始条件以及不光滑的右手边都可能导致不光滑的解。我们为低正则性解决方案提出了新的先验和后验误差估计。为了避免在执行均匀网格细化时出现的收敛速度降低，我们还考虑了基于残差后验误差指标和功能后验误差估计量的自适应细化程序。然后通过时空代数多重网格预处理的 GMRES 求解庞大的时空有限元方程组。特别是在 4d 时空情况下，同时时空并行化可以大大减少计算时间。我们展示并讨论了几个具有不同规律性特征的例子的数值结果。