Two new error estimates of a fully discrete primal-dual mixed finite element scheme for parabolic equations in any space dimension
type: Item
instance of: scholarly article
title: Two new error estimates of a fully discrete primal-dual mixed finite element scheme for parabolic equations in any space dimension
zbMATH Open document ID: 1505.65260
author: Abdallah Bradji
author: Fayssal Benkhaldoun
published in: Results in Mathematics
publication date: 2021-08-19T00:00:00Z
review text: A fully discrete implicit scheme is considered, in which the discretization in space is performed using the Primal-Dual Mixed Finite Element Method (PDMFEM) as an approximation in space for the non-stationary heat equation as a model of parabolic equations in any space dimension. The time discretization is performed using a uniform mesh. The existence and uniqueness of the discrete solution are shown. The convergence results which state the optimal estimate for the error are proved. The organization of this paper is as follows. In Section 1, the problem to be solved, preliminaries, formulation of the PDMFE scheme, and aim of this paper are given. In Section 2, the main convergence results in a general setting of finite element spaces are studied. Section 3 is devoted to the application of the results of Section 2 to the particular case when the mixed finite element spaces are defined by the known Raviart-Thomas spaces of order l. In Section 5, the authors summarize the results of the paper and sketch some perspectives to work on in the near future.
Mathematics Subject Classification ID: 65M60
Mathematics Subject Classification ID: 65M06
Mathematics Subject Classification ID: 65N30
Mathematics Subject Classification ID: 65M15
Mathematics Subject Classification ID: 65M12
Mathematics Subject Classification ID: 35K05
Mathematics Subject Classification ID: 35Q79
zbMATH DE Number: 7383832
zbMATH Keywords: parabolic equation
zbMATH Keywords: mixed finite element method
zbMATH Keywords: new error estimates
zbMATH Keywords: Raviart-Thomas spaces
reviewed by: Temur Jangveladze
MaRDI profile type: Publication
full work available at URL: s00025-021-01489-0
OpenAlex ID: W3196193042
cites work: A new error estimate for a fully finite element discretization scheme for parabolic equations using Crank-Nicolson method.
cites work: Some abstract error estimates of a finite volume scheme for a nonstationary heat equation on general nonconforming multidimensional spatial meshes.
cites work: Finite Elements
cites work: The finite element methods for elliptic problems.
cites work: Parabolic finite volume element equations in nonconvex polygonal domains
cites work: Error estimates for a finite volume element method for parabolic equations in convex polygonal domains
cites work: Global Estimates for Mixed Methods for Second Order Elliptic Equations
cites work: The gradient discretisation method
cites work: Superconvergence for rectangular mixed finite elements
cites work: Mixed finite elements, compatibility conditions, and applications. Lectures given at the C.I.M.E. summer school, Cetraro, Italy, June 26--July 1, 2006
cites work: Adaptive Finite Element Methods for Parabolic Problems VI: Analytic Semigroups
cites work: Adaptive Finite Element Methods for Parabolic Problems V: Long-Time Integration
cites work: Adaptive Finite Element Methods for Parabolic Problems IV: Nonlinear Problems
cites work: Adaptive Finite Element Methods for Parabolic Problems II: Optimal Error Estimates in $L_\infty L_2 $ and $L_\infty L_\infty $
cites work: Adaptive Finite Element Methods for Parabolic Problems I: A Linear Model Problem
cites work: Finite volume methods
cites work: Q4399897
cites work: Q5699647
cites work: Error estimates for some mixed finite element methods for parabolic type problems
cites work: Mixed finite element methods for general quadrilateral grids
cites work: Discrete Maximal Parabolic Regularity for Galerkin Finite Element Methods for Nonautonomous Parabolic Problems
cites work: AnH1-Galerkin Mixed Finite Element Method for Parabolic Partial Differential Equations
cites work: Q3521017
cites work: A priori error estimates of expanded mixed FEM for Kirchhoff type parabolic equation
cites work: Galerkin Finite Element Methods for Parabolic Problems
cites work: Superconvergence analysis of the lowest order rectangular Raviart-Thomas element for semilinear parabolic equation
DOI: 10.1007/S00025-021-01489-0
description: scientific article; zbMATH DE number 7383832