Abstract

In this paper, we design a new framework of direct radial basis function partition of unity (D-RBF-PU) method to solve parabolic equation on surface with and without boundary. Resort to the tangent plane approach, the proposed method avoids dealing with complex surface differentiation operators, and only needs to approximate the standard differentiation operators on the two-dimensional tangent space compared with the existing RBF-PU methods on surface. The new meshfree method is called the "D-RBF-PU tangent plane" method. Additionally, for the surface parabolic equation with Neumann boundary condition, the ghost node technique is considered to avoid the concentration of test nodes on one side of the patch near the boundary and improve the computational accuracy. Numerical examples are performed to confirm the convergence and the eigenvalue stability and applications to the Fitzhugh–Nagumo model and Schnakenberg model on different surfaces demonstrate the efficiency of the proposed method.