A posteriori error estimates and an adaptive scheme of least‐squares meshfree method

Abstract

SUMMARYA posteriori error estimates and an adaptive reÿnement scheme of ÿrst-order least-squares meshfree method (LSMFM) are presented. The error indicators are readily computed from the residual. For an elliptic problem, the error indicators are further improved by applying the Aubin-Nitsche method. It is demonstrated, through numerical examples, that the error indicators coherently re ect the actual error. In the proposed reÿnement scheme, Voronoi cells are used for inserting new nodes at appropriate positions. Numerical examples show that the adaptive ÿrst-order LSMFM, which combines the proposed error indicators and nodal reÿnement scheme, is e ectively applied to the localized problems such as the shock formation in uid dynamics.

The least-squares meshfree method for the steady incompressible viscous flow

“…Here, general results on the convergence characteristics of the first-order least-squares formulation are given. The details can be found in References [31][32][33].…”

“…This implies that simple schemes for generating integration points can be used without degrading the solution accuracy. The convergences under inaccurate integration and an adaptive scheme with a posteriori error estimates for LSMFM have been studied [24,25]. However, the works with LSMFM has been limited to linear problems such as Poisson equation, and thus its application to various engineering fields should be investigated.…”

A three-dimensional adaptive analysis using the meshfree node-based smoothed point interpolation method (NS-PIM)

“…Rabczuk and Belytschko [30] proposed adaptive analysis for structured meshfree particle methods in 2-D and 3-D problems. Park and Kwon [31] used a least square mesh free method with Voronoi cells to refine interesting regions. Angulo et al [32] implemented adaptive produce of meshfree finite point method for solving boundary value problems.…”

An Error Estimator and Adaptivity Based on Basis Coefficient-Vector Field of Element-Free Galerkin Method

“…The two broad methods for computing errors are residual method by Park et al [2003] and recovery method by Chung and Belytschko [1998]. Residual defines the amount with which solution fails to satisfy the governing equation.…”