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A novel high-order approximate scheme for two-dimensional time-fractional diffusion equations with variable coefficient. (English) Zbl 1442.65281

Summary: Based on the spatial quasi-Wilson nonconforming finite element method and temporal \(L2-1_\sigma\) formula, a fully-discrete approximate scheme is proposed for a two-dimensional time-fractional diffusion equations with variable coefficient on anisotropic meshes. In order to demonstrate the stable analysis and error estimates, several lemmas are provided, which focus on high accuracy about projection and superclose estimate between the interpolation and projection. Unconditionally stable analysis are derived in \(L^2\)-norm and broken \(H^1\)-norm. Moreover, convergence result of accuracy \(O(h^2+\tau^2)\) and superclose property of accuracy \(O(h^2+\tau^2)\) are deduced by combining interpolation with projection, where \(h\) and \(\tau\) are the step sizes in space and time, respectively. And then, the global superconvergence is presented by employing interpolation postprocessing operator. Finally, numerical results are provided to demonstrate the validity of the theoretical analysis.

MSC:

65M60 Finite element, Rayleigh-Ritz and Galerkin 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
35R11 Fractional partial differential equations
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