Problems in mathematical foundations of adaptive finite element methods

后验误差估计和自适应有限元方法是在科学和工程应用中广泛使用的求解偏微分方程组的工具。后验估计提供了离散化误差的可计算范围，而AFEM是一种有效的求解技术，通过自动局部网格分级准确地反映了解的性质。这个项目的目标是更好地理解AFEM的数学基础，并在几个特定的应用领域提供新的后验误差估计和自适应算法。该项目的主要部分致力于开发和分析曲面上偏微分方程组的后验误差估计和AFEM。具体项目涉及演化曲面上抛物型偏微分方程组的欧拉公式，仅有离散近似信息的曲面上椭圆型偏微分方程组的解，以及椭圆型特征值问题。另一个重点是有限元的优良性质，特别是在非标准范数下的先验和后验误差估计的发展。PI将在实践中常见的高等级网格类型的这种范数下发展新的先验误差估计，证明椭圆界面问题的新的后验最大范数界，并将类似的误差分析整合到他对曲面特征值问题的研究中。在科学和工程中的广泛应用导致了偏微分方程(PDE)的产生，为了获得对物理世界的准确预测，必须求解偏微分方程(PDE)。在现代应用中，偏微分方程组通常是在计算机上近似求解的，在近似解的质量和计算资源的投入之间存在权衡。PI将研究自适应算法的数学基础，该算法在有效利用手头的计算能力的同时自动生成更准确的解。该项目的一部分旨在丰富对现有算法的数学理解，另一部分旨在为各种应用开发新的、在数学上合理的自适应算法。

A posteriori error estimates and adaptive finite element methods (AFEM) are widely-used tools for solving partial differential equations (PDEs) arising in science and engineering applications. A posteriori estimates provide computable bounds on discretization errors, while AFEM are efficient solution techniques which accurately reflect solution properties via automatic local mesh grading. The goals of this project are to better understand the mathematical underpinnings of AFEM and to provide new a posteriori error estimates and adaptive algorithms in several specific application areas. A major part of the project is devoted to development and analysis of a posteriori error estimates and AFEM for PDEs on surfaces. Specific projects concern Eulerian formulations of parabolic PDEs on evolving surfaces, solution of elliptic PDEs on surfaces for which the only available information is a discrete approximation, and elliptic eigenvalue problems. Another emphasis is fine properties of FEM, in particular the development of a priori and a posteriori error estimates in nonstandard norms. The PI will develop new a priori error estimates in such norms on the types of highly graded meshes typically seen in practice, prove new a posteriori maximum-norm bounds for elliptic interface problems, and integrate similar error analysis into his study of surface eigenvalue problems. A wide variety of applications in science and engineering give rise to partial differential equations (PDEs) which must be solved in order to obtain accurate predictions about the physical world. PDEs are typically solved approximately on computers in modern applications, and there is a tradeoff between the quality of the approximate solution and the investment of computational resources. The PI will study the mathematical underpinnings of adaptive algorithms which automatically generate more accurate solutions while efficiently employing the computing power at hand. Part of the project is aimed at enriching mathematical understanding of existing algorithms, and part to developing new and mathematically well-justified adaptive algorithms for various applications.