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**A posteriori error approximation in EFG method.**
*(English)*
Zbl 1047.74081

Summary: Recently, considerable effort has been devoted to the development of the so-called meshless methods. Meshless methods still require considerable improvement before they equal the prominence of finite elements in computer science and engineering. One of the paths in the evolution of meshless methods has been the development of the element free Galerkin (EFG) method. In the EFG method, it is obviously important that the a posteriori error should be approximated.

An a posteriori error approximation based on the moving least-squares method is proposed, using the solution, computed from the EFG method. The error approximation procedure proposed in this paper is simple to construct and requires, at most, nearest neighbour information from the EFG solution. The formulation is based on employing different moving least-squares approximations. Different selection strategies of the moving least-squares approximations have been used and compared, to obtain optimum values of the parameters involved in the approximation of the error.

The performance of the developed approximation of the error is illustrated by analysing different examples for two-dimensional (2D) potential and elasticity problems, using regular and irregular clouds of points. The implemented procedure of error approximation allows the global energy norm error to be estimated and also provides a good evaluation of local errors.

An a posteriori error approximation based on the moving least-squares method is proposed, using the solution, computed from the EFG method. The error approximation procedure proposed in this paper is simple to construct and requires, at most, nearest neighbour information from the EFG solution. The formulation is based on employing different moving least-squares approximations. Different selection strategies of the moving least-squares approximations have been used and compared, to obtain optimum values of the parameters involved in the approximation of the error.

The performance of the developed approximation of the error is illustrated by analysing different examples for two-dimensional (2D) potential and elasticity problems, using regular and irregular clouds of points. The implemented procedure of error approximation allows the global energy norm error to be estimated and also provides a good evaluation of local errors.

### MSC:

74S30 | Other numerical methods in solid mechanics (MSC2010) |

65N15 | Error bounds for boundary value problems involving PDEs |

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\textit{L. Gavete} et al., Int. J. Numer. Methods Eng. 58, No. 15, 2239--2263 (2003; Zbl 1047.74081)

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