Adaptive Ch Method with Local Coupled Multiquadrics for Solving Partial Differential Equations

TL;DR

This paper introduces an adaptive meshless collocation method using Local Coupled Multiquadrics for efficiently solving partial differential equations with high accuracy and flexibility in local refinement.

Contribution

It proposes the Adaptive Ch Method that automatically adjusts local cover sizes and nodal spacing, enhancing solution accuracy without the need for mesh connectivity.

Findings

Accurate solutions for 1D and 2D Poisson problems

Method is meshless and requires no element connectivity

Effective across various shape parameter values

Abstract

We present a new adaptive collocation scheme for solving partial differential equations based on Local Coupled Multiquadrics (LCMQs) within a covers-and-nodes framework. The method, referred to as the Adaptive Ch Method, automatically prioritizes adjusting the local cover size C then refines local nodal spacing h to achieve a prescribed tolerance. Numerical examples for one- and two-dimensional Poisson problems demonstrate accurate solutions across a wide range of shape parameter values, while preserving the advantages of local collocation. The proposed approximation approach is truly meshless, requiring no element, connectivity or continuity to construct trial functions or weights.