迭代方法用于k-赫森方程
Iterative methods for k-Hessian equations
AI总结：
本文研究了k-赫森方程的迭代方法，分析了九点有限差分方案的局部唯一解和二次收敛性，并提出适用于非光滑解的新迭代方法，与流行的Gauss-Seidel方法建立联系，引入了2-赫森方程的新Gauss-Seidel类型迭代方法。
摘要：
在n维欧几里得空间的域上，对于整数k=1,...,n，k-赫森方程是 fully nonlinear 椭圆方程，当k>1时，k=1时为泊松方程，k=n时为蒙特-安培方程。我们分析了光滑非退化解的九点有限差分方案。证明了离散方案具有局部唯一解和二次收敛率。此外，我们提出适用于非光滑解的新迭代方法。后者与流行的蒙特-安培方程的Gauss-Seidel方法建立联系，并引入了2-赫森方程的新Gauss-Seidel类型迭代方法。
英文摘要：
On a domain of the n-dimensional Euclidean space, and for an integer k=1,...,n, the k-Hessian equations are fully nonlinear elliptic equations for k >1 and consist of the Poisson equation for k=1 and the Monge-Ampere equation for k=n. We analyze for smooth non degenerate solutions a 9-point finite difference scheme. We prove that the discrete scheme has a locally unique solution with a quadratic convergence rate. In addition we propose new iterative methods which are numerically shown to work for non smooth solutions. A connection of the latter with a popular Gauss-Seidel method for the Monge-Ampere equation is established and new Gauss-Seidel type iterative methods for 2-Hessian equations are introduced.
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