Meshfree Collocation Methods for PDEs

• Meshfree collocation methods are numerical techniques for discretizing differential equations without a structured grid, relying on scattered nodes and local basis functions.
• They use local Taylor expansions, radial basis functions, and weighted least squares to approximate function values and their derivatives.
• These methods enable flexible simulation on irregular and high-dimensional domains, enhancing accuracy and computational efficiency.

Meshfree collocation methods are a family of numerical techniques for the discretization of partial differential equations (PDEs) and related operators on unstructured, irregular, or scattered collections of points, without relying on a background mesh. These methods construct discrete approximations of functions and their derivatives by enforcing reproduction conditions based on local Taylor expansions, radial basis function (RBF) interpolants, kernel reproducing Hilbert spaces, or weighted least squares principles. The haLLMark of meshfree collocation is the absence of a fixed computational grid; instead, function values and their required derivatives are approximated directly at arbitrarily placed collocation points—often called particles or nodes—through locally defined basis functions, with accuracy and stability controlled by the choice of reproduction order, neighbor selection, and basis construction.

1. Core Principles and Mathematical Foundations

Meshfree collocation methods approximate a target function $u$ and its derivatives at a discrete set of points $\{x_i\}$ by imposing polynomial reproduction or kernel-based interpolation. Many methods are unified by the following core construction:

• Starting with a local Taylor expansion at $x_i$,

$$u(x_j) = u(x_i) + (x_j - x_i)\cdot\nabla u(x_i) + \frac{1}{2}(x_j-x_i)^T D^2 u(x_i) (x_j-x_i) + \cdots$$

derivatives at $x_i$ (e.g., $\partial^\alpha u(x_i)$) are approximated by summing neighbor values weighted to satisfy “moment” constraints,

$$L^{(\alpha)}u(x_i) = \sum_{j\in\mathcal{N}_i} u_j [W_{ji}\cdot(M_i^{-1}C^{(\alpha)})],$$

where $W_{ji} = X_{ji}\varphi_{ji}$ (with $X_{ji}$ containing local monomials, $\varphi_{ji}$ a chosen weight or kernel), $M_i$ is the local moment matrix, and $C^{(\alpha)}$ selects the appropriate derivative.

• Depending on whether the considered method is based on reproducing kernels (RKHS), radial basis functions, moving least squares (MLS), or generalized finite difference (GFDM) ideas, the structure of $W_{ji}$ and $M_i$ adapts, but the unifying framework is identical (Halada et al., 24 Sep 2025).

This general form accommodates methods with and without explicit interpolation of the function itself: some enforce the reproduction of the zeroth moment (function values), while others focus solely on derivative reproduction.

2. Historical Evolution and Taxonomy

The evolution of meshfree collocation began in the 1970s and 1980s with generalized finite difference schemes on unstructured points, early meshless SPH (smoothed particle hydrodynamics), and later with the development of reproducing kernel particle methods (RKPM) and high-order consistent SPH (HOCSPH). Moving least squares (MLS)—and its generalized (GMLS), compact (CMLS), and interpolating (IMLS) variants—became widespread for their flexibility and accuracy. Finite point methods (FPM), least squares kinetic upwind methods (LSKUM), DC-PSE (discretization-corrected particle strength exchange), and peridynamics-inspired formulations diversified the landscape. Recent works have clarified that many methods, including RBF approaches, can be classified by the strategy of basis construction (Taylor monomials vs. orthogonal polynomials; kernel- vs. polynomial-based), enforcement of the reproduction conditions, and inclusion or exclusion of certain moments (Halada et al., 24 Sep 2025).

The taxonomy organizes methods by their principle of derivation:

• RBF-based collocation (global and local, classical Kansa, Hermite schemes)
• Kernel-based and RKHS (reproducing kernel, generalized interpolation)
• MLS and its variants (GMLS, CMLS, MMLS, etc.)
• GFDM families and moment-consistent approaches
• Physically motivated variants (e.g., peridynamics, DC-PSE)

3. Algorithmic Structure and Generalized Derivation

The generalized derivation begins with the Taylor expansion and requires constructing weights $w_{ji}^{(\alpha)}$ such that the discrete approximation

$$\sum_{j\in\mathcal{N}_i} u(x_j) w_{ji}^{(\alpha)} \approx \partial^\alpha u(x_i)$$

holds for all polynomials up to a desired degree on the discrete stencil. The corresponding reproducing conditions are:

$$\sum_{j\in\mathcal{N}_i} X_{ji}^\beta w_{ji}^{(\alpha)} = C^{(\alpha)}_\beta,$$

over multi-indices $\beta$ in the chosen basis. For anisotropic basis functions, $W_{ji} = X_{ji}\varphi_{ji}$, the moment matrix $M_i$ is invertible if sufficient neighbors and a linearly independent basis are chosen. The discrete operator thus assumes the unifying form:

$$L^{(\alpha)} u(x_i) = \sum_{j\in\mathcal{N}_i} u(x_j) [W_{ji} \cdot (M_i^{-1} C^{(\alpha)})].$$

Variance across methods comes from the structure of $X_{ji}$, choice of weights $\varphi_{ji}$ (e.g., kernel shape, RBF, MLS weight), and specifics like polynomial basis or kernel expansion (Halada et al., 24 Sep 2025).

4. Representative Methods and Comparative Features

The breadth of meshfree collocation is evident in the following techniques:

Method	Basis Functions	Notable Features
RBF Collocation	RBFs	Global/local support, kernel selection, spectral accuracy possible (Nakano, 2014, Garmanjani et al., 2018)
RKHS Collocation	Reproducing Kernel	Cardinal functions, error estimates in RK-norms (Azarnavid et al., 2017, Leng et al., 2019)
MLS/GMLS	Polynomials, Weights	Generalizes SPH/MLS, optimal approximation in Sobolev spaces (Halada et al., 24 Sep 2025, Wenzel et al., 2022)
GFDM/FPM	Taylor monomials	Weighted ℓ₂ minimization for derivative recovery, generalized FD (Kraus et al., 2022)
DC-PSE	Symmetric kernel	Moment condition enforcement for finite difference consistency, extended to surfaces (Singh et al., 2022)
HOLMES	Max-ent basis	High-order, strong-form collocation without integration grid (Greco et al., 2020)

Choice of method is influenced by geometry (e.g., surface vs. volume), required order of consistency, computational cost, and conditioning.

5. Challenges, Stability, and High-Dimensionality

Meshfree collocation schemes are subject to stability and conditioning challenges, especially in the context of large or “flat” basis functions (e.g., small RBF shape parameter), irregular point arrangements, and enforcement of boundary conditions. Various localizations (partition of unity, compact supports), adaptive refinement (Cavoretto et al., 2018), and hybridization (e.g., conservative/strong-form switching in GFDM for interface problems (Kraus et al., 2022)) address these issues. In high dimensions, careful point selection (e.g., PDE-greedy kernel placement (Wenzel et al., 2022)) and target-data dependent strategies can offer dimension-independent convergence rates, mitigating the curse of dimensionality.

Crucially, for singular integral operators (e.g., Lévy-type PIDEs), meshfree collocation with suitably smooth (strictly positive-definite) functions can regularize inherent singularities, bypassing penalties used in standard mesh-based schemes (Damircheli et al., 2021).

6. Applications and Broader Impact

Meshfree collocation methods have been applied across a range of domains:

• Fully nonlinear parabolic equations in viscosity solution frameworks (Nakano, 2014)
• Structural credit risk modeling involving singular PIDEs (Damircheli et al., 2021)
• Surface PDEs and geometric flows on manifolds (Suchde, 2020, Singh et al., 2022)
• Cardiac electrophysiology on complex domains (Mountris et al., 2021)
• Volume-conserving Lagrangian flows (Suchde et al., 2023)
• Adaptive high-resolution solvers for elliptic PDEs and moving interfaces (Cavoretto et al., 2018, Kraus et al., 2022)
• Meshfree spectral and high-order (HOLMES) solvers with direct CAD integration (Greco et al., 2020)

Their meshless nature makes them especially attractive for problems involving moving boundaries, irregular/geometric complexity, high-dimensionality, and coupling with data-driven or adaptive node placement algorithms (Wenzel et al., 2022).

7. Unified Perspectives and Future Directions

A central insight, highlighted by unifying formulations (Halada et al., 24 Sep 2025), is that the vast range of meshfree collocation methods—across historical and disciplinary boundaries—are fundamentally governed by enforcement of polynomial (or kernel) reproduction through local moment conditions and neighbor-weighted stencils, abstracted via a moment matrix $M_i$ and mapping vector $C^{(\alpha)}$. This realization enables cross-transfer of stability, accuracy, and computational strategies between frameworks.

Current and future developments include:

• Systematic exploitation of this unified structure for optimal basis and neighbor selection,
• Hybrid strong/conservative forms for challenging physics (e.g., discontinuous coefficients, volume conservation),
• Adaptive schemes informed by rigorous error indicators and data-driven point placement,
• Efficient cross-validation (for parameter tuning, e.g., RBF shape) to automate solver configuration (Marchetti, 2023),
• Advanced applications to PDEs on manifolds and coupled surface-bulk problems,
• Further integration with high-dimensional and irregular data/modalities, including applications in computational biology, finance, and geophysical flows.

As iterative improvements and extensions emerge, the core principles of meshfree collocation—local point-based reproduction, flexibility of geometry, and tailored enforcement of differential properties—remain central to ongoing advances in meshless numerical simulation and scientific computing.