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arXiv · 2405.11339

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

Shantiram Mahata·Neela Nataraj·Jean-Pierre Raymond
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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.

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Shantiram Mahata, Neela Nataraj, Jean-Pierre Raymond. 2024-05-18. Lowest-order Nonstandard Finite Element Methods for Time-Fractional Biharmonic Problem. https://arxiv.org/abs/2405.11339

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