Our method is non-conforming and uses a localized Lagrange basis that is constructed out of radial basis functions. We use the discrete solution to approximate the continuous solution. We introduce a meshless method for solving both continuous and discrete variational formulations of a volume constrained, non-local diffusion problem. HTML articles powered by AMS MathViewer by The developed meshless method outperforms the classical finite difference method in terms of accuracy in all situations except immediately after the Dirichlet jump where the approximation properties appear similar.A meshless Galerkin method for non-local diffusion using localized kernel bases The accuracy of the method is assessed in terms of the average and maximum errors with respect to the density of nodes, number of nodes in the domain of influence, multiquadrics free parameter, and timestep length on uniform and nonuniform node arrangements. The second is the initial value problem, associated with the Dirichlet jump problem on a square. The first is the boundary value problem (NAFEMS test) associated with the steady temperature field with simultaneous involvement of the Dirichlet, Neumann and Robin boundary conditions on a rectangle. The developed approach thus overcomes the principal large-scale problem bottleneck of the original Kansa method. The computational effort thus grows roughly linearly with the number of the nodes. Only small systems of linear equations with the dimension of the number of nodes included in the domain of influence have to be solved for each node. Instead of global, the collocation is made locally over a set of overlapping domains of influence and the time-stepping is performed in an explicit way. The method is structured on multiquadrics radial basis functions. The formulation copes with the diffusion equation, applicable in the solution of a broad spectrum of scientific and engineering problems. This paper formulates a simple explicit local version of the classical meshless radial basis function collocation (Kansa) method.
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