论文标题
$ hp $ $ hp $ - 抛物线PDE的时空离散量的指数收敛性
Exponential Convergence of $hp$-Time-Stepping in Space-Time Discretizations of Parabolic PDEs
论文作者
论文摘要
对于线性抛物面的初始有限价值问题，具有自动化的，时间均匀的椭圆形空间操作员，以发散形式具有Lipschitz-连续系数，以及在多边形/多面体域中不兼容的，不兼容的，时间分析性的强迫术语，我们在解决方案的时间 - 分析性的时间 - 均值时间。时间分析性通过加权，分析功能类别，有限，空间规律性低并且没有边界兼容性的数据进行量化。利用这一结果，我们证明了一种符合的半混凝土$ HP $ -Time-Steppping方法的指数融合。我们将这种半散布的时间与一阶，所谓的“ $ h $ version”拉格朗日有限元相结合，并在太空中的角转化与张量产品，并符合时空配方的离散化。我们证明，在$ d $的适当角和角边的网状网络中，错误与时空自由度的数量基本上是基本上（达到对数术语），这是标准FEM在$ d $中提供的一个椭圆形边界价值问题解决方案。我们专注于二维空间域，并评论一维情况和三维情况。
For linear parabolic initial-boundary value problems with self-adjoint, time-homogeneous elliptic spatial operator in divergence form with Lipschitz-continuous coefficients, and for incompatible, time-analytic forcing term in polygonal/polyhedral domains $D$, we prove time-analyticity of solutions. Temporal analyticity is quantified in terms of weighted, analytic function classes, for data with finite, low spatial regularity and without boundary compatibility. Leveraging this result, we prove exponential convergence of a conforming, semi-discrete $hp$-time-stepping approach. We combine this semi-discretization in time with first-order, so-called "$h$-version" Lagrangian Finite Elements with corner-refinements in space into a tensor-product, conforming discretization of a space-time formulation. We prove that, under appropriate corner- and corner-edge mesh-refinement of $D$, error vs. number of degrees of freedom in space-time behaves essentially (up to logarithmic terms), to what standard FEM provide for one elliptic boundary value problem solve in $D$. We focus on two-dimensional spatial domains and comment on the one- and the three-dimensional case.
