A meshfree method with a non-overlapping domain decomposition method based on TPS for solving the forward-backward heat equation in two-dimension

We present a truly meshless method based on the thin plate splines for the numerical solution of the two dimensional forward-backward heat equation and give a robust formulation for the proposed method. The physical domain is divided into two subdomains each of which defines a forward or a backward subproblem. The resulting subproblems are treated by a radial basis function method for spatial dimension and a finite difference scheme for the time derivative followed by an iterative domain decomposition method to achieve a desired accuracy. In addition, we propose a combined use of an interpolation and the collocation method, in the iterations to update the interface boundary solution. Furthermore, we show that the time discretization scheme is unconditionally stable and convergent. Finally, some numerical examples will be presented to demonstrate the efficiency of the proposed method and some computational aspects will be discussed.