Finite Element Approximation of Time-Fractional Fourth-Order Problem with Nonlocal Diffusion: Existence–Uniqueness and Error Bounds

Key Points

The existence–uniqueness of the weak solution is established through the Faedo–Galerkin method, highlighting theoretical importance.
This study utilizes a transformation into a system of two second-order equations, enhancing the problem's manageability.
A fully discrete scheme combines finite element method with a specific scheme on a graded mesh, facilitating detailed approximation.
Numerical experiments provide additional validation for the theoretical findings, emphasizing the method’s practical applicability.

Abstract

In this paper, we consider a time-fractional fourth-order nonlocal problem with Navier boundary conditions. First, we discuss the existence–uniqueness of the weak solution at the continuous level using Faedo–Galerkin method. Then this fourth-order problem is transformed into a system of two second-order equations. For this system, a fully discrete scheme is proposed which comprises the standard finite element method and the Formula: see text scheme on the graded mesh. For the proposed scheme, we derive Formula: see text-robust convergence estimates. Finally, numerical experiments are presented to validate the theoretical findings.

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Finite Element Approximation of Time-Fractional Fourth-Order Problem with Nonlocal Diffusion: Existence–Uniqueness and Error Bounds

Key Points

Abstract

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