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Finite element error analysis of a time-fractional nonlocal diffusion equation with the Dirichlet energy. (English) Zbl 1446.65116
Summary: A time-fractional diffusion equation involving the Dirichlet energy is considered with nonlocal diffusion operator in the space which has dimension \(d\in\{2,3\}\) and the Caputo sense fractional derivative in time. Further, nonlocal term in diffusion operator is of Kirchhoff type. We discretize the space using the Galerkin finite elements and time using the finite difference scheme on a uniform mesh. First, we prove the existence and uniqueness of a fully discrete numerical solution of the problem using the Brouwer fixed point theorem. Then, we give a priori bounds and convergence estimates in \(L^2\) and \(L^\infty\) norms for fully-discrete problem. A more delicate analysis in the error provides the second order convergence for the proposed scheme. Numerical results are provided to validate the theoretical analysis.
MSC:
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
35R11 Fractional partial differential equations
26A33 Fractional derivatives and integrals
35B45 A priori estimates in context of PDEs
74H10 Analytic approximation of solutions (perturbation methods, asymptotic methods, series, etc.) of dynamical problems in solid mechanics
Software:
FreeFem++; UMFPACK
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