Topics in Mathematical Theory of Adaptive Finite Element Methods
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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.