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

Ahmed E. Seleit

TL;DR

This work develops an adaptive, meshless solver for PDEs that combines Local Coupled Multiquadrics with a covers-and-nodes framework. The Adaptive $\mathcal{C}h$ method automatically allocates local support via per-node cover sizes $\mathcal{C}$ and refines sampling with nodal insertions $h$, guided by a residual-based a posteriori indicator and a classifier that balances enrichment versus refinement. The method yields sparse, banded global systems and demonstrates high accuracy on 1D and 2D Poisson problems with reduced sensitivity to the shape parameter $c$ compared to standard MQ approaches. The approach offers a practical, scalable path for truly meshless PDE solving with targeted local refinement and stable conditioning.

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.

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

TL;DR

This work develops an adaptive, meshless solver for PDEs that combines Local Coupled Multiquadrics with a covers-and-nodes framework. The Adaptive method automatically allocates local support via per-node cover sizes and refines sampling with nodal insertions , guided by a residual-based a posteriori indicator and a classifier that balances enrichment versus refinement. The method yields sparse, banded global systems and demonstrates high accuracy on 1D and 2D Poisson problems with reduced sensitivity to the shape parameter compared to standard MQ approaches. The approach offers a practical, scalable path for truly meshless PDE solving with targeted local refinement and stable conditioning.

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.

Paper Structure

This paper contains 14 sections, 16 equations, 4 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. LCMQ Covers
  3. Local support
  4. Adaptive Method on LCMQ Covers
  5. Algorithmic rationale and flow.
  6. Initialization (Alg. 1, lines 1--2).
  7. Solve with current covers (line 4).
  8. A classifier to choose between $\mathcal{C}$-Enrichment and $h$-refinement (line 7).
  9. Choosing the dominant action at cycle level (lines 8--9 and 18).
  10. Local $\mathcal{C}$-enrichment and selective $h$-refinement (lines 10--17).
  11. $h$-refinement branch (lines 19--23).
  12. Enforcing caps, limits, and stopping (lines 24--27).
  13. Numerical Experiments
  14. Conclusions

Figures (4)

  • Figure 1: Results for the 1D Poisson problem with adaptive $\mathcal{C}h$--LCMQ.
  • Figure 2: Cover size distribution for the 1D Poisson problem at the last adaptive cycle
  • Figure 3: Numerical solution and absolute error of the 2D Poisson problem.
  • Figure 4: Error in the 2D Poisson problem.