Unconditionally optimal error analysis of Crank-Nicolson Galerkin FEMs for a strongly nonlinear parabolic system
type: Item
instance of: scholarly article
title: Unconditionally optimal error analysis of Crank-Nicolson Galerkin FEMs for a strongly nonlinear parabolic system
zbMATH Open document ID: 1377.65118
author: Dongfang Li
author: Jilu Wang
published in: Journal of Scientific Computing
publication date: 2017-11-10T00:00:00Z
review text: This paper is concerned with the unconditionally optimal error estimates of linearized Crank-Nicolson Galerkin finite element methods (FEMs) for a strongly nonlinear parabolic system \[ c(x,u) u_t-\nabla \cdot (a(x,u)\nabla u)=f(x,u)\quad\text{ in }\Omega\times (0,T], \] subject to boundry condition \(u(x,t)=0\) on \(\partial\Omega\times (0,T]\) and initial condition \(u(x,0)=u_0(x)\) in \(\Omega\times \{0\}\). The approach is developed by the authors in two steps. Firstly, a temporal-spatial error splitting argument is employed in order to derive \(L^2\) error estimates under the condition \(\tau \geq h\) (here \(\tau\) is the time-step size and \(h\) is the spatial mesh size). Secondly, boundedness of numerical solutions are deduced by an inductive process when \(\tau \leq h\). Finally, optimal \(L^2\) and \(H^1\) error estimates are deduced. Numerical results are presented to support the theoretical findings.
reviewed by: Marius Ghergu
Mathematics Subject Classification ID: 65M15
Mathematics Subject Classification ID: 65M60
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 65M06
zbMATH DE Number: 6805231
zbMATH Keywords: unconditionally optimal error analysis
zbMATH Keywords: strongly nonlinear parabolic system
zbMATH Keywords: linearized Crank-Nicolson Galerkin finite element methods
zbMATH Keywords: numerical result
MaRDI profile type: Publication
full work available at URL: s10915-017-0381-3
OpenAlex ID: W2588904077
cites work: Non-classical \(H^ 1\) projection and Galerkin methods for non-linear parabolic integro-differential equations
cites work: A Priori $L^2 $ Error Estimates for Finite-Element Methods for Nonlinear Diffusion Equations with Memory
cites work: Exact solutions to nonlinear diffusion equations obtained by the decomposition method
cites work: Q5189317
cites work: Convergence analysis of an approximation of miscible displacement in porous media by mixed finite elements and a modified method of characteristics
cites work: Finite element method for solving nonlinear parabolic equations
cites work: Optimal Error Estimates of Linearized Crank-Nicolson Galerkin FEMs for the Time-Dependent Ginzburg--Landau Equations in Superconductivity
cites work: Improved error estimates for mixed finite‐element approximations for nonlinear parabolic equations: The discrete‐time case
cites work: A splitting method for incompressible flows with variable density based on a pressure Poisson equation
cites work: A Modified Backward Time Discretization for Nonlinear Parabolic Equations Using Patch Approximations
cites work: Solution to strongly nonlinear parabolic problems by a linear approximation scheme
cites work: Unconditionally Optimal Error Estimates of a Crank--Nicolson Galerkin Method for the Nonlinear Thermistor Equations
cites work: Error analysis of linearized semi-implicit Galerkin finite element methods for nonlinear parabolic equations
cites work: Unconditional convergence and optimal error estimates of a Galerkin-mixed FEM for incompressible miscible flow in porous media
cites work: Implicit-explicit predictor-corrector schemes for nonlinear parabolic differential equations
cites work: A linear finite difference scheme for generalized time fractional Burgers equation
cites work: A new fractional time-stepping method for variable density incompressible flows
cites work: Convergence of Numerical Approximations of the Incompressible Navier–Stokes Equations with Variable Density and Viscosity
cites work: A Galerkin Method for Nonlinear Parabolic Equations with Nonlinear Boundary Conditions
cites work: Q5547721
cites work: Exact solutions to nonlinear diffusion equations obtained by a generalized conditional symmetry method
cites work: Two-Level Discrete-Time Galerkin Approximations for Second Order Nonlinear Parabolic Partial Differential Equations
cites work: Some Optimal Error Estimates for Piecewise Linear Finite Element Approximations
cites work: Unconditional stability and error estimates of modified characteristics FEMs for the Navier-Stokes equations
cites work: Finite difference methods for a nonlinear and strongly coupled heat and moisture transport system in textile materials
cites work: Galerkin method for a nonlinear parabolic–elliptic system with nonlinear mixed boundary conditions
cites work: Q4360234
cites work: A new error analysis of Crank-Nicolson Galerkin FEMs for a generalized nonlinear Schrödinger equation
cites work: Heat and sweat transport in fibrous media with radiation
cites work: A Priori $L_2 $ Error Estimates for Galerkin Approximations to Parabolic Partial Differential Equations
cites work: Curved Elements in the Finite Element Method. I
DOI: 10.1007/S10915-017-0381-3
description: scientific article; zbMATH DE number 6805231