论文标题
关于Monge-Ampère操作员非线性特征值问题的数值解决方案
On the Numerical Solution of Nonlinear Eigenvalue Problems for the Monge-Ampère Operator
论文作者
论文摘要
在本文中，我们报告了在研究Monge-Ampère操作员$ v \ rightarrow \ det \ det \ mathbf {d}^2 v $的某些非线性特征值问题的数值解决方案时获得的结果。我们采用的方法依赖于以下成分：（i）考虑到所考虑的特征值问题的差异。 （ii）通过操作员分解初始值问题（一种梯度流）与每个特征值问题相关的时间离散化。 （iii）依靠连续分段仿射函数空间的有限元近似。为了验证上述方法，我们将其应用于具有已知确切解决方案的问题的解决方案：我们获得的结果表明，当空间离散化步骤$ h \ rightarrow 0 $时，我们获得的结果表明将收敛到精确解决方案。我们还考虑了没有已知确切解决方案的测试问题。
In this article, we report the results we obtained when investigating the numerical solution of some nonlinear eigenvalue problems for the Monge-Ampère operator $v\rightarrow \det \mathbf{D}^2 v$. The methodology we employ relies on the following ingredients: (i) A divergence formulation of the eigenvalue problems under consideration. (ii) The time discretization by operator-splitting of an initial value problem (a kind of gradient flow) associated with each eigenvalue problem. (iii) A finite element approximation relying on spaces of continuous piecewise affine functions. To validate the above methodology, we applied it to the solution of problems with known exact solutions: The results we obtained suggest convergence to the exact solution when the space discretization step $h\rightarrow 0$. We considered also test problems with no known exact solutions.
