Adaptive FEM for elliptic and parabolic problems

该项目的目标是开发和分析各种椭圆和抛物型偏微分方程的自适应有限元方法（AFEM）。该项目有三个主要部分。首先是对控制非标准范数的AFEM收敛性质及其相关应用的理论研究。仅在过去几年，控制全球能源规范的自适应方法的准最优性结果才出现。研究者已经开发了类似的分析AFEM的技术，旨在控制其他规范。他将运用这些技术来证明控制局部能量规范的AFEM的最优性，并使用它们来分析Bank-Holst型并行自适应算法的收敛性。在第二组项目中，研究者将开发AFEM，以有效地控制几类抛物型问题的局部和全局最大误差，其中这种误差控制是可取的。这些努力将产生新的后验误差估计和自适应有限元方法，用于控制线性抛物问题中的局部点误差，以及对此类估计具有实际意义的半线性抛物问题的最大范数后验误差分析。第三个项目涉及开发求解椭圆型和抛物型PDE的AFEM。研究人员将开发一种用于固定问题的表面AFEM，这将有助于解决表面和体效应耦合的问题。他还将研究线性抛物面PDE，以提供关于演化曲面的AFEM的基础知识。偏微分方程在科学和工程中广泛应用于模拟各种物理现象。为了从这些数学模型中获得有用的信息，有必要使用数值（计算机）算法来近似它们的解。自适应有限元法是一种利用可用信息自动有效地提高逼近解质量的数值算法。研究者的研究涉及这种自适应算法的数学理论。这个项目有三个主要目标。首先是研究有关这些算法收敛性的基本理论问题，即证明它们是正确的。该项目的第二部分涉及控制某些自适应计算中的最大误差，也就是说，确保计算在任何地方都是正确的，而不仅仅是标准的“平均”。最后，许多重要的物理模型都涉及到曲面上的偏微分方程。例子包括旋转制动鼓上的油分散和多相流体（如油和水）中表面张力的影响。研究者还将开发自适应算法来解决这些方程。

The goal of this project is the development and analysis of adaptive finite element methods (AFEM) for various elliptic and parabolic partial differential equations. The project has three main parts. The first involves theoretical investigation of convergence properties of AFEM for controlling non-standard norms and related applications. It is only in the past few years that quasi-optimality results for adaptive methods for controlling global energy norms have appeared. The investigator has developed techniques for similarly analyzing AFEM designed to control other norms. He will apply these techniques to prove optimality of an AFEM for controlling local energy norms and use them to analyze convergence of parallel adaptive algorithms of Bank-Holst type. In the second set of projects the investigator will develop AFEM for efficiently controlling local and global maximum errors in several classes of parabolic problems where such error control is desirable. These efforts will result in sharp new a posteriori error estimates and adaptive finite element methods for controlling local pointwise errors in linear parabolic problems and maximum-norm a posteriori error analyses of semilinear parabolic problems for which such estimates are of practical interest. The third project involves development of AFEM for solving elliptic and parabolic PDE on surfaces. The investigator will develop a surface AFEM for stationary problems which will be useful for problems in which surface and bulk effects are coupled. He will also study linear parabolic surface PDE in order to provide foundational knowledge about AFEM on evolving surfaces.Partial differential equations are widely used in science and engineering in order to model various physical phenomena. In order to gain usable information from these mathematical models, it is necessary to approximate their solutions using numerical (computer) algorithms. Adaptive finite element methods are numerical algorithms that use available information in order to automatically and efficiently improve the quality of the approximation to the solution. The investigator's research concerns the mathematical theory of such adaptive algorithms. This project has three main goals. The first is to investigate basic theoretical questions concerning convergence of these algorithms, that is, to prove that they work correctly. The second part of the project involves controlling maximum errors in certain adaptive calculations, that is, ensuring that the computation is correct everywhere and not just "on the average" as is standard. Finally, many important physical models involve partial differential equations on surfaces. Examples include oil dispersion on a rotating brake drum and the effects of surface tension in fluid flows with multiple phases, such as oil and water. The investigator will also develop adaptive algorithms for solving such equations.