论文标题
非自治抛物线问题的自适应时空有限元方法与分配源
Adaptive space-time finite element methods for non-autonomous parabolic problems with distributional sources
论文作者
论文摘要
我们考虑在完全非结构化的简单时空网格上进行局部稳定，符合有限元方案，用于抛物线抛物面初始边界值问题的数值解决方案，具有可变的，可能在时空和时间上不连续的系数，系数。还接纳了分销来源。不连续的系数，非平滑边界，变化的边界条件，非平滑或不兼容的初始条件以及非平滑的右侧可能导致非平滑溶液。我们提出了针对低规度解决方案的新的先验和后验错误估计。为了避免在均匀网格精炼的情况下出现降低的收敛速率，我们还考虑基于残留的自适应精炼程序，后验错误指标和功能性A后验误差估计器。然后，通过时空代数多机预处理的GMRE来求解巨大的时空有限元方程系统。特别是，在空间3D的4D时空情况下，同时时空并行化可以大大减少计算时间。我们介绍并讨论具有不同规律性特征的几个示例的数值结果。
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, possibly discontinuous in space and time, coefficients. 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 convergence rates appearing in the case of 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 that is 3d in space, simultaneous space-time parallelization can considerably reduce the computational time. We present and discuss numerical results for several examples possessing different regularity features.
