Generalized finite differences for solving 3D elliptic and parabolic equations

• The GFDM can be applied for solving problems defined over irregular clouds of points.
• The study of the influence of the main parameters involved in the approximation is performed.
• The treatment of the Neumann boundary conditions.
• The measure of the irregularity of the cloud of points
• A new limit for stability.

The generalized finite difference method (GFDM) is a meshfree method that can be applied for solving problems defined over irregular clouds of points. The GFDM uses the Taylor series development and the moving least squares approximation to obtain explicit formulae for the partial derivatives.In this paper, this meshfree method is used for solving elliptic and parabolic partial differential equations in 3-D. The influence of the main parameters involved in the approximation and the treatment of the Neumann boundary condition are shown. Parabolic equations have been solved using an explicit method and the criterion for stability has been improved taking into account the irregularity of the cloud of points. The numerical results show the high accuracy obtained.