Besov regularity of parabolic differential equations on Lipschitz domains

在这个项目中，我们研究了有界Lipschitz区域上的抛物型偏微分方程解。我们的目的是证明在处理这类方程时使用自适应数值方法是合理的。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。附加自由度仅用于数值近似距离精确解仍很远的区域。从实现最佳N项逼近(即，通过至多N个基函数的线性组合对解的最佳逼近)的收敛速度的意义上来说，自适应算法所能期望的最佳性能是最优性能。然而，这种收敛顺序取决于解在Besov空间的特定尺度下的正则性。因此，我们的目的是研究抛物型偏微分方程解的Besov正则性，以了解自适应在这种情况下是否有效。在该项目的第一个资助期，我们能够证明，对于相当一般的线性和非线性抛物型偏微分方程组，适应性确实是合理的。对于多面体锥体(而不是一般的Lipschitz域)，可以获得更好的正则性结果。在该项目的第二个资助期，我们希望进一步改进和发展这些成果。我们在圆锥方面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，非凸域是特别重要的，我们想在这里证明类似的结果。此外，我们还计划研究随机抛物型偏微分方程组在分数阶Sobolev空间中的正则性，这决定了非自适应方法的收敛阶。此外，还打算研究偏微分方程组在更一般的流形(如肥皂膜)上的Besov正则性。另一方面，我们研究抛物型偏微分方程解的逼近类。我们的目标是当我们使用Galerkin方法进行时间离散和空间自适应离散时，对水平线法(Rothe方法)进行收敛分析。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，这种方法的效率更高。特别地，这种方法可以得到的逼近阶与空间尺度无关，并且依赖于精确解在Besov空间张量积的特定尺度下的正则性。因此，我们将系统地研究支配混合光滑空间在这些尺度上解的正则性。

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.