Improved Meshless Methods Solving Equation With Laplacian Operator

Mesh-based method,for instance finite element method,succeed used in engineering problems,has many limitation.The mesh interpolation based on in low quality brings low accuracy.Moreover some problems is discontinuous that mesh is not available.Some improved approximations built weak formation to achieve continuity which enrich the approximation spaces.Meshless method refrain from above-mentioned problems because it dose not need meshes.However,meshless method is not without limitations that it could be better if it is modified.There are three meshless methods,Trefftz method,Fundamental solution method,and Particular solution method involved in this paper,which divided into two parts.The first part introduces method of fundamental solutions(MFS)and the Collocation Trefftz method which have been known as two highly effective boundary-type methods for solving homogeneous equations.Despite many attractive features of these two methods,they also experience different aspects of difficulty.Recent advances in the selection of source location of the MFS and the techniques in reducing the condition number of the Trefftz method have made significant improvement in the performance of these two methods which have been proven to be theoretically equivalent.In this paper we will compare the numerical performance of these two methods under various smoothness of the boundary and boundary conditions.The topic of second part is that improved particular solution method which adopting polynomial as basis,solving asymmetric equation in three dimension.A transformation of governing equation from 3D to 2D is made to reduce complexity and enhance accuracy during the approximation.Furthermore,we adopt a new approach to approximate the solution by utilizing the polynomial solution corresponding forcing term instead of using polynomial directly.We will compare result of Kansa's-RBF by solving the symmetric poisson equations in 3D with the result of these directly method and indirectly method solving the same symmetric equation after transform the dimension to 2D.