Mesh free analysis with Galerkin finite block method for linear PDEs

Based on the Garlerkin method, the Galerkin finite block method (GFBM) is proposed to deal with two-dimensional (2D) linear partial differential equations (PDEs) with variable coefficients in this paper. The mapping technique is utilized to transform a block in physical domain into normalized square. Physical variables are approximated with double layer Chebyshev polynomials for 2D problem. A set of linear algebraic equation is formulated with the Chebyshev polynomials from PDE and boundary conditions in weak form. Continuous conditions at interfacial surfaces between two blocks are introduced in either weak form or strong form. It is demonstrated that the GFBM is suitable to deal with complicated problems with high accuracy including discontinuous boundary values problem and concentrated heat sources in the domain. Several numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method.