Superconvergence of the Gradient in Piecewise Linear Finite-Element Approximation to a Parabolic Problem

1989 ◽  
Vol 26 (3) ◽  
pp. 553-573 ◽  
Author(s):  
Vidar Thomée ◽  
Jin-Chao Xu ◽  
Nai-Ying Zhang
Keyword(s):  
Finite Element ◽  
Parabolic Problem ◽  
Piecewise Linear ◽  
Finite Element Approximation ◽  
Element Approximation ◽  
Linear Finite Element

FINITE ELEMENT APPROXIMATION OF A MODEL FOR PHASE SEPARATION OF A MULTI-COMPONENT ALLOY WITH NONSMOOTH FREE ENERGY AND A CONCENTRATION DEPENDENT MOBILITY MATRIX

1999 ◽  
Vol 09 (05) ◽  
pp. 627-663 ◽  
Author(s):  
JOHN W. BARRETT ◽  
JAMES F. BLOWEY
Keyword(s):  
Finite Element ◽  
Free Energy ◽  
Phase Separation ◽  
Space Dimension ◽  
Piecewise Linear ◽  
Finite Element Approximation ◽  
Element Approximation ◽  
Linear Finite Element ◽  
Mobility Matrix ◽  
Two Space Dimensions

We consider a model for phase separation of a multi-component alloy with nonsmooth free energy and a concentration dependent mobility matrix. In particular we prove that there exists a unique solution for sufficiently smooth initial data. Further, we prove an error bound for a fully practical piecewise linear finite element approximation in one and two space dimensions. Finally numerical experiments with three components in one space dimension are presented.

Numerical Analysis for a Nonlocal Parabolic Problem

2016 ◽  
Vol 6 (4) ◽  
pp. 434-447 ◽  
Author(s):  
M. Mbehou ◽  
R. Maritz ◽  
P.M.D. Tchepmo
Keyword(s):  
Finite Element ◽  
Parabolic Problem ◽  
Reaction Diffusion ◽  
Finite Element Approximation ◽  
Reaction Diffusion Equation ◽  
Element Approximation ◽  
Galerkin Finite Element ◽  
Fully Discrete ◽  
Nonlinear Reaction Diffusion Equation ◽  
Nonlinear Parabolic

AbstractThis article is devoted to the study of the finite element approximation for a nonlocal nonlinear parabolic problem. Using a linearised Crank-Nicolson Galerkin finite element method for a nonlinear reaction-diffusion equation, we establish the convergence and error bound for the fully discrete scheme. Moreover, important results on exponential decay and vanishing of the solutions in finite time are presented. Finally, some numerical simulations are presented to illustrate our theoretical analysis.

scholarly journals AN IMPROVED FINITE ELEMENT APPROXIMATION AND SUPERCONVERGENCE FOR TEMPERATURE CONTROL PROBLEMS

2013 ◽  
Vol 18 (5) ◽  
pp. 631-640 ◽  
Author(s):  
Yuelong Tang
Keyword(s):  
Finite Element ◽  
Temperature Control ◽  
Piecewise Linear ◽  
A Priori ◽  
Finite Element Approximation ◽  
Linear Functions ◽  
Control Problems ◽  
Element Approximation ◽  
Adjoint State ◽  
Admissible Controls

In this paper, we consider an improved finite element approximation for temperature control problems, where the state and the adjoint state are discretized by piecewise linear functions while the control is not discretized directly. The numerical solution of the control is obtained by a projection of the adjoint state to the set of admissible controls. We derive a priori error estimates and superconvergence of second-order. Moreover, we present some numerical examples to illustrate our theoretical results.

scholarly journals A Two-Grid Method for Finite Element Solutions of Nonlinear Parabolic Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Chuanjun Chen ◽  
Wei Liu
Keyword(s):  
Finite Element ◽  
Piecewise Linear ◽  
A Priori ◽  
Grid Method ◽  
Finite Element Approximation ◽  
Nonlinear Parabolic Equations ◽  
Coarse Grid ◽  
Element Approximation ◽  
Fine Grid ◽  
Nonlinear Parabolic

A two-grid method is presented and discussed for a finite element approximation to a nonlinear parabolic equation in two space dimensions. Piecewise linear trial functions are used. In this two-grid scheme, the full nonlinear problem is solved only on a coarse grid with grid sizeH. The nonlinearities are expanded about the coarse grid solution on a fine gird of sizeh, and the resulting linear system is solved on the fine grid. A priori error estimates are derived with theH1-normO(h+H2)which shows that the two-grid method achieves asymptotically optimal approximation as long as the mesh sizes satisfyh=O(H2). An example is also given to illustrate the theoretical results.

Sign in / Sign up

Export Citation Format

Share Document