The method of approximate fundamental solutions (MAFS) for elliptic equations of general type with variable coefficients

Abstract

The paper presents a meshless method for solving elliptic equations of general type with variable coefficients. It is based on the use of the delta-shaped functions and the method of approximate fundamental solutions first suggested for solving equations with constant coefficients. The method assumes that the solution domain is embedded in a square and the initial equation is extended onto the square with the help of the CICE −(Chebyshev interpolation + C-expansion) approximation scheme. As a result the coefficients of the equation are approximated by the truncated Fourier series over some orthogonal system in the square. The approximate fundamental solutions (AFSs) satisfy L[u]=I(x), where I(x) is the delta shaped function in the form of the truncated Fourier series. Thus, the AFSs due to the special form of the operator can be obtained in the similar form of truncated series. The next part of the MAFS follows the general scheme of the MFS. The numerical examples are presented and the results are compared with the analytical solutions. The comparison shows that the method presented provides a very high precision in solution of two-dimensional elliptic equations of general type with different boundary conditions (Dirichlet, Neumann, mixed) in arbitrary domains.