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求解偏微分方程的常用工具。 后验估计提供了可计算的离散化误差的界限，而AFEM是有效的解决方案技术，通过自动局部网格分级准确地反映解决方案的属性。 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。 该项目的一个主要部分是致力于开发和分析的后验误差估计和AFEM的偏微分方程的表面。 具体项目涉及欧拉公式的抛物型偏微分方程的发展表面上，椭圆型偏微分方程的解决方案的表面上，唯一可用的信息是一个离散近似，椭圆特征值问题。 另一个重点是有限元法的优良性质，特别是在非标准规范的先验和后验误差估计的发展。 PI将开发新的先验误差估计在这种规范的类型的高等级网格通常在实践中看到，证明新的后验最大范数边界椭圆接口问题，并集成类似的误差分析到他的研究表面特征值问题。在科学和工程中的广泛应用产生了偏微分方程（PDE），必须求解这些方程才能获得关于物理世界的准确预测。 在现代应用中，偏微分方程通常在计算机上近似求解，并且在近似解的质量和计算资源的投资之间存在折衷。 PI将研究自适应算法的数学基础，这些算法可以自动生成更精确的解决方案，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.