\( \alpha \)-robust error analysis of a mixed finite element method for a time-fractional biharmonic equation
type: Item
instance of: scholarly article
title: \( \alpha \)-robust error analysis of a mixed finite element method for a time-fractional biharmonic equation
zbMATH Open document ID: 1481.65184
author: Chaobao Huang
author: Martin Stynes
published in: Numerical Algorithms
publication date: 2021-07-23T00:00:00Z
review text: The authors consider the numerical resolution of the linear time fractional biharmonic equation \({D}_t^{\alpha } u+{\Delta }^2u-c{\Delta } u =f\). The fractional derivative is given in the sense of Caputo and its order is satisfying \(\alpha \in (0,1)\). The boundary conditions are \(u = \Delta u = 0\) on \(\partial \Omega \). Some a priori estimates on the exact solution are derived. To perform a numerical scheme, the problem is written under the form of a system of two problems using the new variable \(p=-\Delta u\). A weak formulation is then derived using this system in which the space of the test functions and the solutions is \(H^1_0\). Then, a finite element method is applied to the system. The discretization in time is performed using the well-known L1 scheme on a graded meshes. To analyze the numerical obtained scheme, a convenient discrete Gronwall inequality is derived and proved. A discrete stability is proved and an error estimate is shown as well. Some numerical tests are presented to support the theoretical results. The article is interesting and it merits to read.
reviewed by: Abdallah Bradji
Mathematics Subject Classification ID: 65M60
Mathematics Subject Classification ID: 65M22
Mathematics Subject Classification ID: 65N30
Mathematics Subject Classification ID: 65M12
Mathematics Subject Classification ID: 65M15
Mathematics Subject Classification ID: 31A30
Mathematics Subject Classification ID: 35B45
Mathematics Subject Classification ID: 26A33
Mathematics Subject Classification ID: 35R11
zbMATH DE Number: 7374238
zbMATH Keywords: time-fractional biharmonic equation
zbMATH Keywords: weak singularity
zbMATH Keywords: mixed finite element method
MaRDI profile type: Publication
full work available at URL: s11075-020-01036-y
OpenAlex ID: W3097159432
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DOI: 10.1007/S11075-020-01036-Y
description: scientific article; zbMATH DE number 7374238