Optimal Convergence Rates for Adaptive Finite Element Techniques

数值模拟是对日益复杂的科学和技术过程进行更深入和更定量的洞察的不可或缺的工具。尽管数字计算设施的能力不断增强，但数值模拟技术在某种程度上是一个薄弱环节。该研究项目旨在开发改进的自适应数值算法，它具有优化分配计算资源的能力--即。自由度--在基于迄今收集的信息的解决过程中。借助自适应求解技术尽可能地节省自由度，同时仍然准确地捕捉感兴趣的结构，仍然是大规模模拟的核心，也是最终进一步推进可计算性前沿的基本前提。虽然大规模科学计算通常发生在高度跨学科的领域，但设计具有严格基础的可证明性能保证的自适应算法是本项目追求的内在数学任务。这个研究项目的许多概念方面为有才华的年轻研究人员提供了独特的机会来开发他们的潜力。这个项目旨在通过一个基于局部误差估计器的从粗到精的过程，通过局部分布自由度的过程来开发和分析hp自适应近似。实现项目目标的挑战有两个主要来源。一方面，应用这种方法的偏微分方程的类型当然非常重要。另一方面，有几个与特定应用无关的基本问题方面，主要是近似论性质的。即使将问题限制在固定的网格细化深度和允许的最大多项式次数，寻找最优度分布与充分局部细化的划分相结合也是一个NP-Hard问题。特别是，当从粗略网格发展到连续细化网格时，看起来很好的学位分配可能会在更晚的阶段产生，以防止出现近乎最佳的结果。因此，解决这些核心近似理论问题并了解它们在多大程度上受到特定类型的偏微分方程的影响是至关重要的。例如，当使用协调方法处理一类重要的椭圆边值问题时，试函数需要是全局连续的，这严重阻碍了由于光滑性污染而进行的局部细化分析，特别是在多变量情况下。解决这些方面的问题，并为未来不同应用领域的规范奠定坚实的基础是本项目的主要目标。一些设想的理论结果预计将是渐近性质的。因此，理论研究将伴随着对模型问题的策略的实施，这些策略揭示了方法的数量行为。高水平的适应性干扰了并行化，开启了另一个研究方向，特别是关于现代处理器技术。

Numerical simulation is an indispensable tool for acquiring deeper and more quantitative insight into increasingly complex scientific and technological processes. Despite the ever-increasing power of digital computing facilities, numerical simulation technology is somewhat of a weak link. This research project aims to develop improved adaptive numerical algorithms, which have the ability to optimally allocate computational resources -- viz. degrees of freedom -- in the course of the solution process based on information gathered so far. Economizing as much as possible the number of degrees of freedom with the aid of adaptive solution techniques while still accurately capturing the structures of interest remains central to large-scale simulation and a fundamental prerequisite for ultimately further advancing the frontiers of computability. While large-scale scientific computation usually takes place in a highly interdisciplinary arena, the design of adaptive algorithms with rigorously-founded certifiable performance guarantees is an inherently mathematical task that is pursued in this project. The many conceptual facets of this research project additionally offer unique opportunities for talented young researchers to develop their potential.This project aims at developing and analyzing hp-adaptive approximations through a process of locally distributing the degrees of freedom through a coarse-to-fine procedure based on local error estimators. The challenges in accomplishing the goals of the project have two major sources. On the one hand, the type of the partial differential equation to which such methods are to be applied, of course, matters very much. On the other hand, there are several fundamental problem aspects that are independent of the particular application and are primarily of approximation-theoretic nature. Even when restricting the problem to a fixed mesh refinement depth and a largest allowable polynomial degree, finding the optimal degree distribution in conjunction with an adequately locally-refined partition is an NP-hard problem. In particular, when progressing from coarse to successively refined meshes seemingly good degree assignments could turn out at a much later stage to prevent near-optimal results. It is therefore of crucial importance to address such core approximation theoretic issues and understand to what extent they are affected by the particular type of partial differential equation. For instance, when using conforming methods for the important class of elliptic boundary value problems, trial functions need to be globally continuous, which severely impedes the analysis of local refinements due to "smoothness pollution," particularly in the multivariate case. To address these aspects and build a solid footing for future specifications to different application areas is the primary goal of this project. Some of the envisaged theoretical results are expected to be of asymptotic nature. Therefore, the theoretical investigations will be accompanied by implementing the strategies for model problems that shed light on the quantitative behavior of the methods. A high level of adaptivity interferes with parallelization, opening yet another direction of research, especially regarding modern processor technologies.