LAMPLEMP

LAMPLEMP, short for Laplacian-augmented Multi-Point Localized Expansion Method, is a numerical technique for solving partial differential equations on irregular domains. It combines localized polynomial expansions with a Laplacian-based stabilization to derive discrete operators without relying on a conventional mesh. The method assigns to each node a small local neighborhood and computes a polynomial approximation of the unknown field that best fits the governing equation and boundary conditions within that neighborhood. The Laplacian operator provides weighting and smoothing of neighboring contributions, improving stability on unstructured point layouts.

Algorithmically, LAMPLEMP proceeds by selecting a stencil around each node, solving a weighted least-squares problem to determine polynomial coefficients that minimize PDE residuals, and incorporating boundary information through ghost points or penalty terms. The resulting local discretizations are assembled into a global sparse system, which is then solved using iterative methods. The approach is compatible with adaptive refinement, higher-order polynomial degrees, and parallel computation.

Advantages of LAMPLEMP include its meshless or mesh-light nature, which eases handling of complex geometries and moving domains. It can achieve high-order accuracy by increasing the local polynomial degree and by enlarging the stencil, while avoiding some mesh-generation costs. Limitations involve sensitivity to stencil quality and conditioning, especially for large-scale problems, and a higher implementation complexity relative to standard finite element methods. Careful selection of weights, boundary treatments, and stabilization strategies are essential to robust performance.

Applications span Poisson and Helmholtz problems, heat conduction, elasticity, and certain computer graphics tasks where irregular geometry or dynamic domains are present. The method is part of the broader family of localized meshless techniques and continues to be explored in numerical analysis literature.