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Stability and convergence of time-stepping methods for a nonlocal model for diffusion

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  • A time-dependent nonlocal model for diffusion is considered. A feature of the model is that instead of boundary conditions, constraints over regions having finite measures are imposed. The explicit forward-Euler, implicit backward-Euler, and Crank-Nicolson methods are considered for discretizing the time derivative and piecewise-linear finite element methods are used for spatial discretization. The unconditional stability of the backward-Euler and Crank-Nicolson schemes and the conditional stability of the forward-Euler scheme are proved as are optimal error estimates for all three schemes. Comparisons with the analogous results for classical local diffusion problems, e.g., the heat equation, are provided as are the results of numerical experiments that illustrate the theoretical results.
    Mathematics Subject Classification: Primary: 65M60, 76R50; Secondary: 45K05.


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