Numerical treatment of singularities and the Generalized Finite Element Method: theory, algorithms, and applications

偏微分方程的数值方法是许多科学和工程领域的重要组成部分。通常在偏微分方程的应用中，人们必须处理由几何、系数或边界条件中的奇点引起的额外困难。本提案的主要目标是为具有奇异解的方程获得更有效的数值方法，并开发用户友好的实现。PI和他的合作者将设计改进的网格，为具有非光滑界面的多面体域上的椭圆方程的三维有限元方法提供最佳的收敛速度。这些方程构成了许多其他应用的基本成分之一。结果将推广到拟线性椭圆方程和演化方程。对于其中一些方程，有时需要具有更高平滑度或其他附加性质的空间，并且将使用广义有限元法来实现这些目的。该提案将通过建议和指导他们以及将他们纳入研究项目来促进研究生和本科生的形成。PI还计划继续举办跨学科研讨会，邀请包括非数学家在内的外部演讲者。提出的研究将导致设计和测试新的和改进的偏微分方程（PDEs）的数值方法在科学和工程领域的兴趣。由此产生的数值方法将以一种非数学家也能理解的方式呈现和实现。它还将通过与其他领域（包括私营部门）的研究人员的互动，有助于将数学结果应用于实践。它也将有助于创造能够使用先进的理论工具来处理实际问题的专家。他还将在更传统的本科课程中引入数值方法和计算机的使用。此外，还计划普及数学家们在实践中提出的数学问题。理论结果和由此产生的数值方法将应用于数学物理，量子化学，生物学，金融（风险管理和金融衍生品定价，或期权）和其他领域。

Numerical Methods for Partial Differential Equations (PDEs) are anessential ingredient in many areas of Sciences and Engineering. Oftenin the applications of PDEs, one has to deal with additionaldifficulties caused by the singularities in the geometry, thecoefficients, or the boundary conditions. The main goals of thisproposal are to obtain more efficient numerical methods for equationswith singular solutions and to develop user friendly implementations.The PI and his collaborators will design improved meshes that provideoptimal rates of convergence for the Finite Element Method in threedimensions for elliptic equations on polyhedral domains withnon-smooth interfaces. These equations form one of the basicingredients in many other applications. The results will be extendedto quasi-linear elliptic equations and to evolution equations. Forsome of these equations, spaces with higher smoothness or other additionalproperties are sometimes needed, and the Generalized Finite ElementMethod will be used for these purposes.The proposal will contribute to the formation of graduate andundergraduate students by advising and mentoring them and byintegrating them in research projects. The PI also plans to continueto run an Interdisciplinary Seminar featuring outside speakers,including non-mathematicians. The proposed research will lead to thedesign and testing of new and improved numerical methods for PartialDifferential Equations (PDEs) of interest in Sciences andEngineering. The resulting numerical methods will be presented andimplemented in a way that makes them accessible tonon-mathematicians. It will also contribute to applying mathematicalresults in practice by interaction with researchers from other fields,including the private sector. It will also contribute to the creationof specialists able to use advanced theoretical tools to handlepractical problems. He will also introduce numerical methods and theuse of computers in more traditional undergraduate courses. The PIalso plans to popularize mathematical questions that arise frompractice among mathematicians. The theoretical results and theresulting numerical methods will have applications in MathematicalPhysics, Quantum Chemistry, Biology, Finance (Risk Management andpricing of Financial Derivatives, or Options), and other areas.