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

几十年来，Monge-Ampere方程和相关模型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主题。然而，如果Monge-Ampere方程的数学和相关模型激励了许多研究人员，并且是大量文献的来源，那么它们的数值就不能说了；当考虑它们的数值解时，这些方程的“完全非线性”很可能是劝阻因素。考虑到这些事实，本项目的目标是：(I)研究求解真实Monge-Ampere方程的计算方法，特别是依赖于混合有限元近似。(Ii)研究求解(I)中离散问题的有效迭代方法；在初步研究的基础上，可以预期快速Poisson求解器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。(3)利用所得到的数值方法，从完全非线性偏微分方程组理论和微分几何出发，如高斯曲率方程，研究了Monge-Ampere型方程的数学性质。(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求解(这类问题发生在流体力学和非线性弹性力学中)。Monge-Ampere方程和相关模型在不同的数学领域中扮演着重要的角色，如微分几何、偏微分方程组和变分。实际上，这类方程也出现在更多的应用领域，如流体力学、非线性弹性力学、材料科学、数学金融等，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，它们所激发的数值工作很少；这些方程的复杂性可能是造成这种矛盾情况的原因。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来求解Monge-Ampere方程和相关的数学问题。将作出系统努力，尽可能依靠“搁置”的现有方法，制定一种模块化的方法。这些研究应该对理论科学和应用科学都有好处；事实上，它们应该：(I)在计算/应用数学和“更具理论性”的数学界之间建立联系和促进合作。(2)让学生(和学院)参与高度跨学科的调查。(3)鼓励数值分析员和计算科学家关注一个重要的跨学科领域，而这一领域显然没有得到充分的研究。(4)教授课程，拓宽学生的知识基础，使他们进入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。

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.