Adaptivity in Local Kernel Based Methods for Approximating Solutions to the Poisson Equation

Expanding on the recent development of adaptive local kernel methods for approximating the action of linear operators, a local estimate of the error and an adaptive procedure for approximating solutions to the Poisson equation is developed.

The error estimate is used in the midst of the adaptive procedure to determine locations in the solution domain where decreasing the spacing between nodes can decrease the error in the solution.

The approach described here is essentially "meshless", with only local Delaunay triangulations leveraged as a convenience to provide information on the local geometry of the node set.

Once this information is utilized, it is discarded.

The experiments performed show close agreement between the error estimate and actual absolute forward error in the approximate solution along with a comparison to existing indicators for refinement.

The combination of the adaptive procedure and error estimate provides an automated technique to resolve localized features of a solution without the, often intractable, expense of uniform refinement across the entire solution domain.