Meshfree collocation solution of boundary value problems via interpolating moving least squares

Abstract

SUMMARYThis paper presents a meshfree interpolating moving least squares (IMLS) method based on singular weights for the solution of partial di erential equations. Due to the speciÿc singular choice of weight functions, which is needed to guarantee the interpolation, there arises a problem of ÿnding the inverse of the occurring singular matrix. The inverse is carried out using a regularization of weight functions. It turns out that a stable inverse is obtained by considering the vanishing regularization parameter.Moreover, the use of this perturbation technique allows the correct evaluation of all necessary derivatives in interpolation points at a reasonable cost.Unlike standard kernel functions used in EFGM, RKPM, etc., the singular kernel functions lead to really interpolating functions which satisfy the Kronecker-delta property. They can be used for enforcement of Dirichlet boundary conditions when solving boundary value problems.Solution to a model BVP as well as an experimental convergence study of the method to analytical solutions, conÿrming the mathematical derivations, are given.

A singularity-avoiding moving least squares scheme for two-dimensional unstructured meshes

“…However, no further discussion is provided of the general conditions under which singularities occur. The singularity problem for moving least squares has also been noted by Netuzhylov [4] and Prax et al [5], and for a similar problem by Schoenauer and Adolph [6], who also observed the occurrence of singularities when the data points lie along straight lines. However, to the best knowledge of the authors, a general theory for predicting the occurrence of singularities in moving least squares has never been published.…”

Space–time meshfree collocation method: Methodology and application to initial‐boundary value problems

“…in the finite differences. We refer the interested reader to [5,20,28] for a detailed description. We will see later on that the initial conditions in the presented approach can be seen as the boundary conditions at the level t = 0.…”

“…A strong-form meshfree method (MCM-IMLS) has been introduced with primary studies in [6,20,21]. In this paper we extend the approach and apply it to time-dependent problems by a consistent discretization of temporal and spacial parts.…”

A new meshless collocation method for partial differential equations

“…The elliptic-type problem has been discretized according to Reference [16], where meshless approximation functions are generated by interpolating moving least-square methods. Poisson's equation in operator notation can be expressed as…”