The Coupling Method And Domain Decomposition Method Based On Boundary Element Method And Meshless Local Petrov-Galerkin Method

The boundary element method (BEM) is a widely used method because of its accuracy and dimensionality reduction. The meshless local Petrov–Galerkin (MLPG) method is a promising meshless method for solving partial differential equations. It is well-suited to problems involving non-homogeneous, anisotropic and non-linear problems.Firstly, the MLPG method and improved MLS (IMLS) approximation are combined, and an improved MLPG (IMLPG) method for two-dimensional analogical Helmholtz equation is discussed in this paper. In the MLPG method, the MLS approximation is used to obtain the approximation function. The algebra equation system in MLS approximation is sometimes ill conditioned. So a new method to establish the approximation function, IMLS approximation, was presented. In IMLS approximation, an orthogonal function system with a weight function is used as the basis function. In comparison with MLS approximation, the algebra equation system in IMLS approximation is not ill conditioned, and can be solved without having to obtain the inverse matrix. In our numerical tests, the numerical convergence of IMLPG method is studied and accurate results are obtained.Secondly, The MLPG method is a genuine meshless method which does not need''elements''or''mesh,''but uses a distributed set of nodes for both field interpolation and background integration. Although the MLPG method has many advantages, the method is more computationally expensive than the finite element method (FEM) and the boundary element method (BEM). So a coupled BEM and MLPG method for analyzing two-dimensional potential problems is presented in this paper. In this method, the analysis domain is divided into two non-overlapping regions (BEM region and MLPG region). The proposed coupling method directly couples the two methods without transition region, the continuity conditions are satisfied on the interface of the two sub-regions. The final system of algebra equation is composed of the BEM equation, the MLPG equation and the equation formed by the continuity conditions. This coupling method is numerically treated under several subdomain models. Some numerical examples for the potential problems governed by the Laplace and Poisson equations are presented to evaluate the accuracy and efficiency of the proposed technique. Thirdly, the coupling method must employs an entire unified equation for the whole domain, by combining the discretized equations for the BEM and MLPG sub-domains, so a non-overlapping domain decomposition algorithm based on the BEM and MLPG method is presented in this paper. This algorithm is iterative in nature. It essentially involves subdivision of the problem domain into subregions being respectively modeled by the two methods, as well as restoration of the original problem with continuity and equilibrium being satisfied along the interface. To speed up the rate at which the algorithm converges, static and dynamic relaxation parameters are employed. The validity of the algorithm is verified by solving some potential problems.