Galerkin approximation of a nonlocal diffusion equation on Euclidean and fractal domains
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The continuum limit of a system of interacting particles on a convergent family of graphs can be described by a nonlocal evolution equation in the limit as the number of particles goes to infinity. Given the continuum limit, the discrete model can be viewed as a Galerkin approximation of the limiting continuous equation. We estimate the speed of convergence of the Galerkin scheme for the model at hand on Euclidean and fractal domains. The latter are relevant when the underlying family of graphs approximates a fractal. Conversely, this paper proposes a Galerkin scheme for a nonlocal diffusion equation on self–similar domains and establishes its convergence rate. Convergence analysis is complemented with numerical integration results for a model problem on Sierpinski Triangle. The rate of convergence of numerical solutions of the model problem fits well the analytical estimate.
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