Abstract
This paper considers some multiscale radial basis function collocation methods for solving the two-dimensional Monge–Ampère equation with Dirichlet boundary. We discuss and study the performance of the three kinds of multiscale methods. The first method is the cascadic meshfree method, which was proposed by Liu and He (2013). The second method is the stationary multilevel method, which was proposed by Floater and Iske (1996), and is used to solve the fully nonlinear partial differential equation in the paper for the first time. The third is the hierarchical radial basis function method, which is constructed by employing successive refinement scattered data sets and scaled compactly supported radial basis functions with varying support radii. Compared with the first two methods, the hierarchical radial basis function method can not only solve the present problem on a single level with higher accuracy and lower computational cost but also produce highly sparse nonlinear discrete system. These observations are obtained by taking the direct approach of numerical experimentation.
Highlights
Introduction e MongeAmpere equation with Dirichlet boundary is a fully nonlinear partial differential equation, which is given by detD2u(z) f(z), in Ω, (1)u(z) g(z), in zΩ, (2)where z has n independent variables in a bounded domain Ω ⊂ Rd and D2u is the Hessian of the function u
Where z has n independent variables in a bounded domain Ω ⊂ Rd and D2u is the Hessian of the function u
In order to design a certain type of numerical method which can avoid the tedious mesh generation and the domain integration and can be capable of dealing with any irregular distribution of nodes, some meshfree collocation methods have been researched by Liu and He [4, 5], Bohmer and Schaback [6], and Rashidinia and collaborators [7,8,9]
Summary
Is paper considers some multiscale radial basis function collocation methods for solving the two-dimensional Monge–Ampere equation with Dirichlet boundary. E second method is the stationary multilevel method, which was proposed by Floater and Iske (1996), and is used to solve the fully nonlinear partial differential equation in the paper for the first time. Compared with the first two methods, the hierarchical radial basis function method can solve the present problem on a single level with higher accuracy and lower computational cost and produce highly sparse nonlinear discrete system. Mathematical Problems in Engineering collocation for the Monge–Ampere equation, we need to design some efficient multiscale radial basis function collocation methods. This is the motivation of the paper.
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