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**A boundary element-free method (BEFM) for two-dimensional potential problems.**
*(English)*
Zbl 1160.65348

Summary: Combining the boundary integral equation (BIE) method and improved moving least-squares (IMLS) approximation, a direct meshless BIE method, which is called the boundary element-free method (BEFM), for two-dimensional potential problems is discussed in this paper. In the IMLS approximation, the weighted orthogonal functions are used as the basis functions; then the system of linear equations is not ill-conditioned and can be solved without obtaining the inverse matrix. Based on the IMLS approximation and the BIE for two-dimensional potential problems, the formulae of the BEFM are given. The BEFM is a direct numerical method in which the basic unknown quantity is the real solution of the nodal variables, and the boundary conditions can be applied directly and easily; thus, it gives a greater computational precision. Some numerical examples are presented to demonstrate the method.

### MSC:

65N38 | Boundary element methods for boundary value problems involving PDEs |

35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |

### Keywords:

moving least-squares (MLS) approximation; improved moving least-squares (IMLS) approximation; weighted orthogonal function; boundary integral equation; meshless method; boundary element-free method (BEFM); potential problems; Poisson equation; numerical examples
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\textit{M. Peng} and \textit{Y. Cheng}, Eng. Anal. Bound. Elem. 33, No. 1, 77--82 (2009; Zbl 1160.65348)

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### References:

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