Optimal adaptive finite element and wavelet methods for p-Poisson equations

本课题主要研究几类拟线性问题的自适应策略的设计，特别是p-Poisson方程的自适应策略的设计、收敛分析以及关于自由度和代数复杂性的最优性证明。我们的方法是基于所谓的Kacanov迭代的自适应正则化，其正则化参数根据具有指导自适应离散的功能的后验误差估计器来调整。我们将同时关注自适应有限元方法和小波方法。动机有两个：一方面，已经为p-Poisson方程定义和研究了适当的有限元离散的可靠误差估计器，我们期望我们能够将这一知识“移植”到小波方法中，对于这个特殊问题，可靠的误差估计器还没有可用的。另一方面，与有限元方法相比，利用小波的强分析性质，通常可以更简单，有时更严格地推导出自适应小波格式的收敛和最优性分析；此外，迄今为止所考虑的任何类型的已知椭圆方程解的Besov正则性的理解都是基于小波的使用。让我们强调这样一个事实，当涉及到自适应有限元和小波方法的收敛速度或复杂性时，解的Besov正则性是一个基本问题。除了分析p-Poisson方程的自适应方法外，我们还计划进行广泛的数值模拟，以证明理论结果的有效性。

This project is concerned with the design of adaptive strategies for certain classes of quasilinear problems, in particular p-Poisson equations, their convergence analysis, and the proof of optimality in terms of the number of degrees of freedom and the algebraic complexity, respectively. Our approach is based on an adaptive regularization of so-called Kacanov iterations, whose regularization parameter is tuned according to a posteriori error estimators which have also the function of guiding adaptive discretizations. We shall focus on both, adaptive finite element and wavelet methods.The motivation is twofold: on the one hand, appropriate reliable error estimators for finite element discretizations have already been defined and studied for the p-Poisson equation, and we expect that we will be able to „port“ this knowledge to wavelet methods for which, in this particular problem, reliable error estimators are not yet available. On the other hand, the strong analytical properties of wavelets can usually be exploited to derive more simply and sometimes more rigorously a convergence and optimality analysis for adaptive wavelet schemes compared to finite element approaches; moreover, the understanding of Besov regularity of solutions of any type of known elliptic equations so far considered has been based on the use of wavelets. Let us stress the fact that Besov regularity of solutions is a fundamental issue when it comes to address the rate of convergence or the complexity of both adaptive finite element and wavelet methods. In addition to the analysis of the adaptive methods for p-Poisson equations, we also plan to perform extensive numerical simulations in order to demonstrate the validity of the theoretical results.