Besov regularity of parabolic differential equations on Lipschitz domains

Lipschitz 域上抛物型微分方程的 Besov 正则

• 批准号: 320243287
• 负责人: Privatdozentin Dr. Cornelia Schneider
• 金额: --
• 依托单位: Department Mathematik
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2016
• 资助国家: 德国
• 起止时间: 2015-12-31 至 2021-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/320243287?language=en
• 关键词: Besov; regularity; parabolic; differential; equations

In this project we study parabolic partial differential equations (=PDEs) on bounded Lipschitz domains. We aim at justifying the use of adaptive numerical methods when treating such equations. In an adaptive strategy the choice of the underlying degrees of freedom is not a priori fixed but depends on the shape of the unknown solutions. Additional degrees of freedom are only spent in regions where the numerical approximation is still far away from the exact solution. The best one can expect from an adaptive algorithm is an optimal performance in the sense that it realizes the convergence rate of best N-term approximation (i.e., best approximation of the solution by linear combinations with at most N basis functions). However, this convergence order depends on the regularity of the solution in specific scales of Besov spaces. It is therefore our aim to investigate the Besov regularity of the solutions of parabolic PDEs in order to see whether adaptivity pays off in this context. In the first funding period of the project we were able to show that adaptivity is indeed justified for quite general classes of linear and nonlinear parabolic PDEs. Even better regularity results could be achieved for polyhedral cones (instead of general Lipschitz domains). In the second funding period of the project we wish to improve and develop these results further. Our achievements for cones have to be generalized to polyhedral domains. Moreover, the nonlinear results on the Besov regularity so far are only established on convex domains. Since from a numerical point of view non-convex domains are of particular interest, we want to prove similar results here. Furthermore, we plan to study the regularity in fractional Sobolev spaces for stochastic parabolic PDEs, which determines the convergence order of non-adaptive methods. Also an investigation of the Besov regularity of PDEs on more general manifolds (e.g. soap films) is intended. As another aspect we study the approximation classes of parabolic PDEs. Our goal here is a convergence analysis of the horizontal mothod of lines (Rothe's method), when we use a Galerkin-method for our discretization in time and adaptive discretizations in space. Moreover, instead of a time-marching algorithm (as described above) we could use a full space-time adaptive algorithm based on tensor wavelets. Numerical studies indicate that this is more efficient. In particular, the approximation order that can be achieved this way turns out to be independent of the spatial dimansion and depends on the regulairity of the exact solution in a specific scale of tensor products of Besov spaces. Therefore, we will systematically investigate the regularity of the solutions in these scales of dominating mixed smoothness spaces.

在这个项目中，我们研究有界Lipschitz域上的抛物型偏微分方程。我们的目的是证明使用自适应数值方法处理此类方程。在自适应策略中，基本自由度的选择不是先验固定的，而是取决于未知解的形状。额外的自由度仅用于数值近似仍然远离精确解的区域。从自适应算法可以期望的最佳性能是在其实现最佳N项近似的收敛速率（即，通过具有最多N个基函数的线性组合的解的最佳逼近）。然而，这种收敛阶依赖于在特定尺度的Besov空间的解的正则性。因此，我们的目标是调查的抛物型偏微分方程的解决方案的Besov正则性，以了解是否自适应支付在这种情况下。在该项目的第一个资助期内，我们能够证明，自适应性确实是合理的相当一般类的线性和非线性抛物型偏微分方程。对于多面体锥（而不是一般的Lipschitz域），甚至可以获得更好的正则性结果。在项目的第二个供资期，我们希望进一步改进和发展这些成果。我们在锥上的成果必须推广到多面体区域。此外，迄今为止关于Besov正则性的非线性结果仅建立在凸域上。由于从数值的角度来看，非凸域是特别感兴趣的，我们想在这里证明类似的结果。此外，我们计划研究分数Sobolev空间中随机抛物型偏微分方程的正则性，它决定了非自适应方法的收敛阶。此外，调查的Besov正则性的偏微分方程更一般的流形（如肥皂膜）的目的。另一方面，我们研究了抛物型偏微分方程的逼近类。我们的目标是在这里是一个收敛性分析的水平线法（Rothe的方法），当我们使用Galerkin方法为我们的离散时间和自适应离散空间。此外，代替时间推进算法（如上所述），我们可以使用基于张量小波的全时空自适应算法。数值研究表明，这是更有效的。特别是，可以实现这种方式的逼近阶原来是独立的空间dimmansion和依赖于在一个特定规模的Besov空间的张量积的精确解的正则性。因此，我们将系统地研究这些尺度的控制混合光滑空间的解的正则性。