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

TL;DR

This paper develops and analyzes lowest-order nonstandard finite element methods for solving time-fractional biharmonic equations, providing optimal error bounds and validating them through numerical experiments.

Contribution

It introduces a unified analysis for several finite element schemes applied to time-fractional biharmonic problems, including error estimates for nonsmooth data.

Findings

Optimal error bounds established for various finite element schemes.

Numerical experiments confirm theoretical convergence rates.

Applicable to both smooth and nonsmooth initial data.

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.