Time second-order finite difference/finite element algorithm for nonlinear time-fractional diffusion problem with fourth-order derivative term. (English) Zbl 1419.65068
Summary: In this article, we study and analyze a Galerkin mixed finite element (MFE) method combined with time second-order discrete scheme for solving nonlinear time fractional diffusion equation with fourth-order derivative term. We firstly introduce an auxiliary variable \(\sigma=\Delta u\), reduce the fourth-order problem into a coupled system with two equations, discretize the obtained coupled system at time \(t_{k-\frac{\alpha}{2}}\) by a second-order difference scheme with second-order approximation for fractional derivative, then formulate mixed weak formulation and fully discrete MFE scheme. Further, we give the detailed proof for stability of scheme, the existence and uniqueness of MFE solution, and a priori error estimates. Finally, by some numerical computations, we test the theoretical results, which illustrate that we can obtain the numerical results for two variables, moreover, we arrive at second-order time convergence orders, which are higher than the ones yielded by the \(L1\)-approximation.
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 |
| 65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |
| 35R11 | Fractional partial differential equations |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
Keywords:
nonlinear fourth-order time fractional diffusion equations; Galerkin mixed finite element algorithm; second-order difference scheme in time; error analysis; stability; existence and uniquenessReferences:
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