Numerical methods for fully nonlinear elliptic equations of the Monge-Ampere type
Numerical methods for fully nonlinear elliptic equations of the Monge-Ampere type
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DOI：
10.1016/j.cma.2005.05.023
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发表时间：
2006-01-01
影响因子：
7.2
通讯作者：
Glowinski, R
中科院分区：
文献类型：
--
作者：
Dean, EJ;Glowinski, R
本文讨论二维Monge-Ampere方程Dirichlet问题的数值解。密切相关的问题的解决方案也进行了讨论，其中包括一个家庭的普奇方程，方程规定的调和平均值的特征值的Hessian的光滑函数的两个变量，和一个最小化问题，从非线性弹性，其中的成本功能涉及的行列式的梯度向量值函数。为了求解Monge-Ampere方程，我们考虑两种方法。第一个“减少”蒙日安培方程的鞍点问题的一个精心选择的增广拉格朗日，解决这个鞍点问题，我们提倡Uzawa道格拉斯Rachford算法。第二种方法结合了非线性最小二乘和算子分裂。第二种方法更容易实现，我们将其变体应用于解决其他问题。对于空间离散化，我们使用混合有限元近似，密切相关的方法已经用于线性和非线性双调和问题的解决方案，通过这些近似的解决上述问题，基本上，减少到离散泊松问题的解决方案。数值实验结果验证了本文所讨论的方法的有效性。(c)2005年由Elsevier B. V.出版
In this article, we discuss the numerical solution of the Dirichlet problem for the Monge-Ampere equation in two dimensions. The solution of closely related problems is also discussed; these include a family of Pucci's equations, the equation prescribing the harmonic mean of the eigenvalues of the Hessian of a smooth function of two variables, and a minimization problem from nonlinear elasticity, where the cost functional involves the determinant of the gradient of vector-valued functions. To solve the Monge-Ampere equation we consider two methods. The first one "reduces" the Monge-Ampere equation to a saddle-point problem for a well-chosen augmented Lagrangian; to solve this saddle-point problem we advocate an Uzawa-Douglas-Rachford algorithm. The second method combines nonlinear least-squares and operator-splitting. This second method being simpler to implements, we apply variants of it to the solution of the other problems. For the space discretization we use mixed finite element approximations, closely related to methods already used for the solution of linear and nonlinear bi-harmonic problems; through these approximations the solution of the above problems is, essentially, reduced to the solution of discrete Poisson problems. The methods discussed in this article are validated by the results of numerical experiments. (c) 2005 Published by Elsevier B.V.