Adaptive meshless methods in electromagnetic modeling: A gradient-based refinement strategy

Meshless methods are numerical methods that have the advantage of high accuracy without the need of an explicitly described mesh topology. In this class of methods, the Radial Point Interpolation Method (RPIM) is a promising collocation method where the application of radial basis functions yields high interpolation accuracy for even strongly unstructured node distributions. For electromagnetic simulations in particular, this distinguishing characteristic translates into an enhanced capability for conformal and multi-scale modeling. The method also facilitates adaptive discretization refinements, which provides an important tool to decrease memory consumption and computation time. In this paper, a refinement strategy is introduced for RPIM. In the proposed node adaptation algorithm, the accuracy of a solution is increased iteratively based on an initial solution with a coarse discretization. In contrast to the commonly used residual-based adaptivity algorithms, this definition is extended by an error estimator based on the solution gradient. In the studied cases this strategy leads to increased convergence rates compared with the standard algorithm. Numerical examples are provided to illustrate the effectiveness of the algorithm.