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

该项目关注的是某些类的拟线性问题的自适应策略的设计，特别是p-Poisson方程，他们的收敛性分析，并证明最优性的自由度和代数复杂性的数量，分别。我们的方法是基于所谓的Kacanov迭代的自适应正则化，其正则化参数根据后验误差估计也有指导自适应离散化的功能进行调整。我们将集中讨论自适应有限元和小波方法，其动机是双重的：一方面，已经为p-Poisson方程定义和研究了有限元离散化的适当可靠的误差估计，我们期望我们能够将这些知识“移植”到小波方法中，在这个特定的问题中，可靠的误差估计还没有。另一方面，小波的强分析性质通常可以被利用，以获得更简单，有时更严格的收敛性和最优性分析的自适应小波计划相比，有限元方法;此外，理解的Besov正则性的解决方案的任何类型的已知椭圆方程迄今被认为是基于小波的使用。让我们强调一个事实，即Besov正则性的解决方案是一个基本问题，当它涉及到解决收敛速度或自适应有限元和小波方法的复杂性。除了分析p-Poisson方程的自适应方法外，我们还计划进行大量的数值模拟，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.