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科学和工程许多领域的重要组成部分。在偏微分方程的应用中，常常要处理由几何奇异性、系数奇异性或边界条件奇异性引起的额外困难。该提案的主要目标是获得更有效的数值方法与奇异解方程和开发用户友好的implementation.The PI和他的合作者将设计改进的网格，提供最佳的收敛速度的有限元方法在三维的椭圆方程多面体域上的非光滑界面。这些方程构成了许多其他应用中的基本要素之一。所得结果将推广到拟线性椭圆型方程和发展方程。对于这些方程中的一些，有时需要具有更高光滑度或其他附加性质的空间，而广义有限元方法将用于这些目的。该建议将有助于研究生和本科生的形成，建议和指导他们，并通过将他们纳入研究项目。PI还计划继续举办一个跨学科研讨会，邀请外部演讲者，包括非数学家。这项研究将为科学和工程领域中的偏微分方程（PDE）设计和测试新的和改进的数值方法提供参考。由此产生的数值方法将提出和实施的方式，使他们获得tonon-mathematicians。它还将通过与包括私营部门在内的其他领域的研究人员的互动，促进将研究成果应用于实践。它还将有助于培养能够使用先进的理论工具来解决实际问题的专家。他还将介绍数值方法和计算机在传统本科课程中的应用。PI还计划在数学家中推广从实践中产生的数学问题。理论结果和由此产生的数值方法将在数学物理、量子化学、生物学、金融（金融衍生品或期权的风险管理和定价）等领域得到应用。

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.