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

TL;DR

LSQD introduces a geometry-free, meshless discretization for elliptic PDEs by representing the solution with local polynomial expansions around scattered nodes and linking neighboring expansions through pointwise PDE and continuity constraints to form a rectangular least-squares system. The method avoids weak formulations and quadrature, enabling straightforward implementation on non-trivial geometries such as quadtree grids while achieving $h$-$p$ convergence. A built-in node-based error estimator provides practical error localization to guide adaptivity, and the approach remains robust even in degenerate, non-coercive, or ill-posed settings. Collectively, LSQD offers a simple, scalable alternative to classical mesh-based methods with strong performance on irregular domains and complex boundary conditions.

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.

Figures (12)

• Figure 1: We represent the solution of a differential equation as a set of local expansions, denoted by $u_i(x)$, which are centered at arbitrary positions $x_{i=1..N}$. We build an overdetermined linear system by evaluating the differential equation around each neighborhood and enforcing continuity conditions between neighboring local representations. The coefficients $\alpha_i^q$ are the least squares solution to the overdetermined system.
• Figure 2: 1D Method - solution representation for increasing spatial and polynomial resolution. We start from a randomly generated set of points $\mathbf{x}_{i=1..10}$ and then recursively split the grid by adding one grid point exactly between each existing pair of direct neighbors. The exact solution is depicted in dashed lines.
• Figure 3: 1D Method - (a) h-P convergence analysis for Poisson's equation with Dirichlet boundary conditions. $L^\infty$ error and corresponding Estimated Order of Accuracy (EOA). (b) Condition number of the least squares matrix $A^TA$. (c) Condition number of the preconditioned least squares matrix, using an incomplete Cholesky decomposition.
• Figure 4: Extension to higher dimensions: domain of interest $\Omega$, and local vicinities $\Omega_i, \Omega_j$. The neighborhoods $V_i, V_j$ are defined as the set of points contained in the corresponding vicinities.
• Figure 5: LSQD Approximation of the function $u_{\textrm{exact}}=\cos(x+y)$ for increasing polynomial order on an arbitrary Quadtree grid.