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Local \(H^1\)-norm error analysis of a mixed finite element method for a time-fractional biharmonic equation. (English) Zbl 1484.65222

Summary: In this work, a time-fractional biharmonic equation with a Caputo derivative of fractional order \(\alpha \in(0, 1)\) is considered, whose solutions exhibit a weak singularity at initial time \(t = 0\). For this problem, a system of two second-order differential equations is derived by introducing a intermediate variable \(p = -\Delta u\), then discretised the system using the standard finite element method in space together with the L1 discretisation of Caputo derivative on graded mesh in time. The \(H^1\)-norm stability result of the method is established, and then a sharp \(H^1\)-norm local convergent result is presented. Finally, numerical experiments are provided to further verify our theoretical analysis for each fixed value of \(\alpha \).

MSC:

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35R11 Fractional partial differential equations
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
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