Lowest-order Nonstandard Finite Element Methods for Time-Fractional Biharmonic Problem

TL;DR

This work addresses a time-fractional biharmonic problem with clamped boundary conditions in a 2D domain, employing lowest-order nonstandard FEMs (Morley, dG, $C^0$IP) for spatial discretization and a graded-time L1 scheme for temporal discretization. A novel Ritz projection $\mathcal{R}_h$ enables a unified energy-based error analysis that handles both smooth and nonsmooth initial data, yielding optimal semidiscrete error bounds in $L^2$ and energy norms with explicit time-singularity factors. The fully discrete analysis is established for smooth data, with convergence rates combining spatial $O(h^{1+\gamma})$ (up to $\gamma_0$) and temporal $O(N^{-{r\alpha}})$-type terms on graded meshes. Numerical experiments corroborate the theoretical rates across Morley, dG, and $C^0$IP schemes for both nonsmooth and smooth initial data, highlighting the method’s robustness and practical relevance for time-fractional fourth-order problems. The results advance finite-element analysis for fractional-in-time fourth-order problems with clamped BCs and open avenues for future work on temporal discretization refinements.

Abstract

In this work, we consider an initial-boundary value problem for a time-fractional biharmonic equation in a bounded polygonal domain with a Lipschitz continuous boundary in $\mathbb{R}^2$ with clamped boundary conditions. After establishing the well-posedness, we focus on some regularity results of the solution with respect to the regularity of the problem data. The spatially semidiscrete scheme covers several popular lowest-order piecewise-quadratic finite element schemes, namely, Morley, discontinuous Galerkin, and $C^0$ interior penalty methods, and includes both smooth and nonsmooth initial data. Optimal order error bounds with respect to the regularity assumptions on the data are proved for both homogeneous and nonhomogeneous problems. The numerical experiments validate the theoretical convergence rate results.

Figures (4)

• Figure 1: Numerical solutions of dG method on $256\times 256$ mesh with $\alpha=0.25,0.5, 0.75$.
• Figure 2: Exact (left) and numerical (right) solutions of dG method on $96\times 96$ mesh, $\alpha=0.5$.
• Figure 3: Exact (left) and numerical (right) solutions of $C^0$IP method on $96\times 96$ mesh, $\alpha=0.5$.
• Figure 4: $L^2$ errors (left) and energy errors (right) plots for case (ii) for the Morley, dG, and $C^0$IP schemes at $t=0.1$ with $k=0.001$

Theorems & Definitions (9)

• Remark 2.1
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