A Meshless Galerkin Method For Non-Local Diffusion Using Localized Kernel Bases
For researchers in numerical methods for non-local diffusion, this provides a novel meshless approach with theoretical guarantees, though it is an incremental improvement over existing meshless methods.
The paper introduces a meshless Galerkin method for non-local diffusion problems using localized kernel bases, demonstrating well-posedness and convergence through inf-sup conditions. Numerical results show a well-conditioned symmetric stiffness matrix.
We introduce a meshless method for solving both continuous and discrete variational formulations of a volume constrained, nonlocal diffusion problem. We use the discrete solution to approximate the continuous solution. Our method is nonconforming and uses a localized Lagrange basis that is constructed out of radial basis functions. By verifying that certain inf-sup conditions hold, we demonstrate that both the continuous and discrete problems are well-posed, and also present numerical and theoretical results for the convergence behavior of the method. The stiffness matrix is assembled by a special quadrature routine unique to the localized basis. Combining the quadrature method with the localized basis produces a well-conditioned, symmetric matrix. This then is used to find the discretized solution.