A Novel Semi-Analytic Mesh less Method for Solving Two- and Three-Dimensional Elliptic Equations of General Form with Variable Coefficients in Irregular Domains

摘要

The paper presents a new meshless numerical method for solving 2D and 3D boundary value problems (BVPs) with elliptic PDEs of general form. The coefficients of the PDEs including the main operator part are spatially dependent functions. The key idea of the method is the use of the basis functions which satisfy the homogeneous boundary conditions of the problem. This allows us to seek an approximate solution in the form which satisfies the boundary conditions of the initial problem with any choice of the free parameters. As a result we separate approximation of the boundary conditions and approximation of the PDE inside the solution domain. Numerical experiments are carried out for accuracy and convergence investigations. A comparison of the numerical results obtained in the paper with the exact solutions and with the data obtained with the use of other numerical techniques (Kansa's method, the method of particular solutions) is performed.