Least SQuares Discretizations (LSQD): a robust and versatile meshless paradigm for solving elliptic PDEs

TL;DR

The paper introduces LSQD, a new meshless method for solving elliptic PDEs that simplifies implementation, handles complex geometries effectively, and demonstrates robust convergence and error estimation.

Contribution

It presents LSQD, a novel meshless discretization approach that avoids weak formulations and quadrature rules, enhancing ease of use and versatility for elliptic PDEs.

Findings

Effective in complex geometries and boundary conditions

Achieves h-P convergence across various parameters

Includes an a posteriori built-in error estimator

Abstract

Searching for numerical methods that combine facility and efficiency, while remaining accurate and versatile, is critical. Often, irregular geometries challenge traditional methods that rely on structured or body-fitted meshes. Meshless methods mitigate these issues but oftentimes require the weak formulation which involves defining quadrature rules over potentially intricate geometries. To overcome these challenges, we propose the Least Squares Discretization (LSQD) method. This novel approach simplifies the application of meshless methods by eliminating the need for a weak formulation and necessitates minimal numerical analysis. It offers significant advantages in terms of ease of implementation and adaptability to complex geometries. In this paper, we demonstrate the efficacy of the LSQD method in solving elliptic partial differential equations for a variety of boundary conditions, geometries, and data layouts. We monitor h-P convergence across these parameters and construct an a posteriori built-in error estimator to establish our method as a robust and accessible numerical alternative.