A Note on a Meshless Method for Fractional Laplacian at Arbitrary Irregular Meshes

The existence and uniqueness of the discrete solutions of a porous medium equation with diffusion are demonstrated. The Cauchy problem contains a fractional Laplacian and it is equivalent to the extension formulation in the sense of trace and harmonic extension operators. By using the generalized finite difference method, we obtain the convergence of the numerical solution to the classical/theoretical solution of the equation for nonnegative initial data sufficiently smooth and bounded. This procedure allows us to use meshes with complicated geometry (more realistic) or with an irregular distribution of nodes (providing more accurate solutions where needed). Some numerical results are presented in arbitrary irregular meshes to illustrate the potential of the method.

Automated summary

The study demonstrates the existence and uniqueness of discrete solutions of a porous medium equation with diffusion, using the generalized finite difference method to achieve convergence of the numerical solution. This method allows for the use of meshes with complicated geometry or irregular node distributions, providing more accurate solutions when dealing with sufficiently smooth and bounded nonnegative initial data.