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

蒙日-安培方程及其相关模型几十年来一直是微分几何学者和非线性偏微分方程专家非常积极研究的课题。然而，如果蒙日-安培方程的数学和相关模型激励了许多研究者，并在一个丰富的文献起源，同样不能说他们的数值；在考虑它们的数值解时，这些方程的“完全非线性”很可能是阻碍因素。考虑到这些事实，本项目的目标是：(i)研究实际蒙日-安培方程的解的计算方法，特别是依靠混合有限元近似。研究有效的迭代方法来解决由第1项导出的离散问题；在初步研究的基础上，可以预期快速泊松解和适当的预条件共轭梯度算法将在求解过程中发挥重要作用。（iii）利用由此产生的数值方法，从完全非线性偏微分方程理论和微分几何（如高斯曲率方程）中研究蒙日-安培型方程的数学性质。（iv）将上述方法应用于涉及蒙日-安培相关模型的自然科学和工程科学问题的数值解（此类问题发生在流体力学和非线性弹性等学科中）。蒙日-安培方程及其相关模型在数学的各个领域，如微分几何、偏微分方程和变分学中发挥着重要作用。实际上，这类方程也出现在流体力学、非线性弹性、材料科学、数学金融学等更多的应用领域，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，尽管上面提到了这些方程的重要性，但与不太基本的模型相比，它们很少激发数值工作；这些方程的复杂性可能是造成这种矛盾局面的原因。这个面向计算的项目的主要目标是为求解蒙日-安培方程和相关数学问题构建用户友好的高效数值方法。将会有一个系统的努力来推导一个模块化的方法，尽可能多地依赖于“现成的”现有方法。这些研究应有利于理论科学和应用科学；事实上，他们应该：(i)在计算/应用数学和“更理论化”的数学界之间建立桥梁和促进合作。（ii）让学生（和教员）参与高度跨学科的研究。鼓励数值分析人员和计算科学家研究一个重要的跨学科领域，这个领域显然没有得到充分的研究。（四）引导课程的教学，拓宽学生的知识基础，并将他们引入一个高度多学科的领域。促进国际合作，因为关于这些题目的合作，特别是同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、会议和专门网站提供。

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.