Problems in mathematical foundations of adaptive finite element methods

后验误差估计和自适应有限元法（AFEM）是在科学和工程应用中广泛使用的求解偏微分方程（PDEs）的工具。后验估计提供了离散误差的可计算边界，而AFEM是有效的求解技术，通过自动局部网格分级准确反映解的性质。该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检误差估计和自适应算法。该项目的一个主要部分是致力于开发和分析表面上偏微分方程的后验误差估计和AFEM。具体项目涉及演化曲面上抛物偏微分方程的欧拉公式，唯一可用信息是离散近似的曲面上椭圆偏微分方程的解，以及椭圆特征值问题。另一个重点是有限元的优良特性，特别是在非标准规范中先验和后验误差估计的发展。PI将在实践中通常看到的高度分级网格类型的这种规范中开发新的先验误差估计，为椭圆界面问题证明新的后验最大规范界限，并将类似的误差分析集成到他对表面特征值问题的研究中。在科学和工程中的广泛应用会产生偏微分方程（PDEs），为了获得对物理世界的准确预测，必须求解偏微分方程。在现代应用程序中，偏微分方程通常在计算机上进行近似求解，并且在近似解的质量和计算资源的投资之间存在权衡。PI将研究自适应算法的数学基础，该算法可以自动生成更准确的解决方案，同时有效地利用手头的计算能力。该项目的一部分目的是丰富对现有算法的数学理解，部分目的是为各种应用开发新的和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.