关于Monge-Ampère算子非线性本征值问题的数值解
On the Numerical Solution of Nonlinear Eigenvalue Problems for the Monge-Ampère Operator
摘要：
本文报道了我们在研究Monge-Ampère算子$v\rightarrow \det \mathbf{D}^2 v$的某些非线性本征值问题的数值解时所得到的结果。我们采用的方法依赖于以下成分：（i）发散制定的本征值问题正在考虑中。(ii)与每个本征值问题相关的初值问题（一类梯度流）的算子分裂时间离散化。(iii)一种依赖于连续分段仿射函数空间的有限元逼近。为了验证上述方法，我们将其应用到已知的精确解的问题的解决方案：我们得到的结果表明收敛到精确解时，空间离散步骤$h\rightarrow 0$。我们还考虑了没有已知精确解的测试问题。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
