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

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.