Cascadic meshfree method for the elliptic Monge–Ampère equation

The elliptic Monge–Ampère equation is a fully nonlinear partial differential equation, which originated in geometric surface theory and has been widely applied in dynamic meteorology, elasticity, geometric optics, image processing and others. The numerical solution of the elliptic Monge–Ampère equation has been a subject of increasing interest recently. In this paper, we design a cascadic algorithm which is meshfree. We first generate hierarchical scattered data sets. Then on each successive refinement levels, the Monge–Ampère equation can be solved by Kansaʹs method. We call this method as cascadic meshfree method (CMF). Different from cascadic multigrid method, CMF avoids tedious interpolation and is more easy for implementation and coding. Finally, numerical experiments confirm the efficiency and robustness of CMF method.