Optimal Convergence Rates for Adaptive Finite Element Techniques

University Of South Carolina At Columbia, Columbia SC

Investigators

Abstract

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.