##
**Meshless analysis of potential problems in three dimensions with the hybrid boundary node method.**
*(English)*
Zbl 1048.65121

Summary: Combining a modified functional with the moving least-squares (MLS) approximation, the hybrid boundary node method (hybrid BNM) is a truly meshless, boundary-only method. The method may have advantages from the meshless local boundary integral equation method and also the boundary node method (BNM). In fact, the hybrid BNN requires only the discrete nodes located on the surface of the domain.

The hybrid BNM has been applied to solve 2D potential problems. In this paper, the hybrid BNM is extended to solve potential problems in three dimensions. Formulations of the hybrid BNM for 3D potential problems and the MLS approximation on a generic surface are developed. A general computer code of the hybrid BNM is implemented in C++. The main drawback of the boundary layer effect in the hybrid BNM in the 2D case is circumvented by an adaptive face integration scheme. The parameters that influence the performance of this method are studied through three different geometries and known analytical fields. Numerical results for the solution of the 3D Laplace’s equation show that high convergence rates with mesh refinement and high accuracy are achievable.

The hybrid BNM has been applied to solve 2D potential problems. In this paper, the hybrid BNM is extended to solve potential problems in three dimensions. Formulations of the hybrid BNM for 3D potential problems and the MLS approximation on a generic surface are developed. A general computer code of the hybrid BNM is implemented in C++. The main drawback of the boundary layer effect in the hybrid BNM in the 2D case is circumvented by an adaptive face integration scheme. The parameters that influence the performance of this method are studied through three different geometries and known analytical fields. Numerical results for the solution of the 3D Laplace’s equation show that high convergence rates with mesh refinement and high accuracy are achievable.

### MSC:

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

65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |

65N50 | Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs |

31B10 | Integral representations, integral operators, integral equations methods in higher dimensions |

31B20 | Boundary value and inverse problems for harmonic functions in higher dimensions |

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