Numerical Methods for Fully Nonlinear Elliptic Equations of the Monge-Ampere Type

Monge-Ampere型完全非线性椭圆方程的数值方法

• 批准号: 0412267
• 负责人: Roland Glowinski
• 金额: --
• 依托单位: University of Houston
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-07-15 至 2008-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0412267&HistoricalAwards=false
• 关键词: Numerical; Methods; Fully; Nonlinear; Elliptic

The Monge-Ampere equation and related models have been for several decades topics of very active investigations by differential geometers and nonlinear partial differential equation specialists. However, if the Mathematics of Monge-Ampere equations and related models have motivated many investigators, and is at the origin of an abundant literature, the same can not be said of their Numerics; it is likely that the "full nonlinearity" of these equations is the dissuasive factor when considering their numerical solution. Taking these facts into account, the objectives of this project are: (i) Investigate computational methods for the solution of real Monge-Ampere equations, relying in particular on mixed finite element approximations. (ii) Investigate efficient iterative methods for the solution of the discrete problems derived from (i); on the basis of preliminary investigations one can expect fast Poisson solvers and properly preconditioned conjugate gradient algorithms to play an important role in the solution process. (iii) Use the resulting numerical methods to investigate the mathematical properties of Monge-Ampere type equations from the theory of fully nonlinear partial differential equations and from Differential Geometry, such as the Gaussian curvature equation. (iv)Apply the above methods to the numerical solution of problems from natural and engineering sciences involving Monge-Ampere related models (such problems take place in, e.g., Fluid Mechanics and Nonlinear Elasticity).The Monge-Ampere equation and related models play an important role in various areas of Mathematics, such as Differential Geometry, Partial Differential Equations and Calculus of Variations. Actually, this type of equations occur also in more applied areas such as Fluid Mechanics, Nonlinear Elasticity, Material Sciences, Mathematical Finance, and from that point of view their numerical solution is an issue of practical importance. Surprisingly, and despite the above mentioned importance of these equations, they have motivated very little numerical work, compared to less fundamental models; the complexity of these equations may be the cause of this paradoxical situation. The main objective of this computationally oriented project is the construction of user friendly efficient numerical methods for the solution of the Monge-Ampere equation and related mathematical problems. There will be a systematic effort to derive a modular methodology, relying as much as possible on "on the shelf" existing methods. These investigations should be beneficial to both the theoretical and applied sciences; indeed, they should: (i) Create bridges and foster cooperation between the computational/applied mathematics and the "more theoretical" mathematics communities. (ii) Involve students (and faculties) in highly interdisciplinary investigations. (iii) Motivate numerical analysts and computational scientists to look at an important interdisciplinary field, which has been clearly under-investigated. (iv)Lead to the teaching of courses broadening the knowledge basis of students and introducing them to a highly multidisciplinary field. (v) Foster international cooperation, since collaborations on these topics, with European scientists in particular, are taking place already. The results of these investigations will be made available via publications, conferences, and dedicated web sites.

Monge-Ampere方程及其相关模型几十年来一直是微分几何学家和非线性偏微分方程专家非常活跃的研究课题。然而，如果数学的蒙赫-安培方程和相关的模型已经激励了许多研究者，是一个丰富的文献的起源，同样不能说他们的数值;它很可能是“完全非线性”的这些方程是劝阻因素时，考虑他们的数值解。考虑到这些事实，本项目的目标是：（一）研究解决真实的Monge-Ampere方程的计算方法，特别是依靠混合有限元近似法。(ii)研究有效的迭代方法来解决来自（i）的离散问题;在初步研究的基础上，人们可以期望快速泊松求解器和适当的预处理共轭梯度算法在求解过程中发挥重要作用。(iii)使用由此产生的数值方法来研究完全非线性偏微分方程理论和微分几何中Monge-Ampere型方程的数学性质，例如高斯曲率方程。（iv）将上述方法应用于涉及Monge-Ampere相关模型的自然科学和工程科学问题的数值解（这样的问题发生在，例如，Monge-Ampere方程及其相关模型在数学的各个领域中发挥着重要作用，例如微分几何、偏微分方程和变分法。实际上，这种类型的方程也出现在更多的应用领域，如流体力学，非线性弹性，材料科学，数学金融，从这个角度来看，它们的数值解是一个具有实际意义的问题。令人惊讶的是，尽管上述这些方程的重要性，他们有动机很少的数值工作相比，不太基本的模型，这些方程的复杂性可能是这种矛盾的情况的原因。这个面向计算的项目的主要目标是构建用户友好的高效数值方法，用于解决Monge-Ampere方程和相关数学问题。将有系统地努力制定一种单元方法，尽可能依靠“现成”的现有方法。这些研究应该有益于理论科学和应用科学;事实上，它们应该：（i）在计算/应用数学和“更理论化”的数学界之间建立桥梁并促进合作。(ii)让学生（和教师）参与高度跨学科的调查。(iii)激励数值分析师和计算科学家关注一个重要的跨学科领域，这显然是研究不足的。（四）导致课程教学扩大学生的知识基础，并向他们介绍一个高度多学科的领域。(v)促进国际合作，因为在这些主题上的合作，特别是与欧洲科学家的合作已经在进行。这些调查的结果将通过出版物、会议和专门网站公布。