Topics in Mathematical Theory of Adaptive Finite Element Methods

有限元法（FEM）被广泛用于近似求解工程和物理科学中出现的物理现象的模拟中的偏微分方程。这种模拟是开发和测试新技术不可或缺的工具。 有限元方法的自适应变体旨在通过更好地利用计算资源来提高模拟的效率和准确性，并通过为研究人员提供近似技术中出现的误差的可计算测量来提高模拟准确性的信心。 该研究项目旨在开发自适应有限元方法的新变体，并增加对其基础的数学理解。 该项目有两个主要重点。 第一种是自适应有限元法，用于定义在表面上的偏微分方程，例如在描述具有多个组分（如油和水）的流体流动时出现。第二个是开发和分析自适应有限元控制各种措施的错误，特别是最大的错误。在第一个项目中，调查员将构建和分析表面有限元方法的自适应变量有两个主要目标。 首先，虽然表面有限元法是一种已建立的有限元方法，定义了许多有用的变量，但缺少一些重要变量的自适应版本。 该项目旨在填补这一空白。 其次，该项目将探索自适应曲面有限元法（有限元代码中表示给定曲面的方式）与曲面的光滑度或规则性之间的相互作用。 其结果将是更强大的自适应表面代码，使用户在表示表面时具有更大的灵活性，同时也尽可能地利用有关表面的可用信息。 第二个主要项目将导致收敛的自适应算法控制最大误差的证明，也将提供新的自适应算法控制最大误差的一类奇摄动椭圆问题。

Finite element methods (FEM) are widely used to approximately solve partial differential equations in simulations of physical phenomena arising in engineering and the physical sciences. Such simulations are an indispensable tool in the development and testing of new technologies. Adaptive variants of finite element methods are designed to increase the efficiency and accuracy with which simulations can be carried out by making better use of computational resources and to increase confidence in the accuracy of simulations by providing researchers with a computable measure of the errors that arise in approximation techniques. This research project aims to develop new variants of adaptive finite element methods and increase mathematical understanding of their underpinnings. The project has two main foci. The first is adaptive FEM for partial differential equations defined on surfaces, which arise for example in describing fluid flows with multiple components (such as oil and water). The second is development and analysis of adaptive FEM for controlling various measures of the error, especially maximum errors.In the first project the investigator will construct and analyze adaptive variants of surface finite element methods with two main goals in mind. First, while surface FEM are an established finite element methodology with many useful variants defined, adaptive versions of some important variants are missing. This project aims to fill that gap. Secondly, the project will explore the interaction between adaptive surface FEM, the way a given surface is represented in a finite element code, and the smoothness or regularity of the surface. The result will be more robust adaptive surface codes that give users greater flexibility in representing surfaces while also making the best possible use of available information about the surface. The second main project will lead to proof of convergence of adaptive algorithms for controlling maximum errors, and will also provide new adaptive algorithms for controlling maximum errors in a class of singularly perturbed elliptic problems.