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.

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 Finite Element Method for a Second-Order Nonlinear Hyperbolic Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Chuanjun Chen ◽  
Wei Liu ◽  
Xin Zhao
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Hyperbolic Equation ◽  
A Priori ◽  
Grid Method ◽  
Second Order ◽  
Coarse Grid ◽  
Nonlinear Hyperbolic Equation ◽  
Fine Grid ◽  
Element Method

We present a two-grid finite element scheme for the approximation of a second-order nonlinear hyperbolic equation in two space dimensions. In the two-grid scheme, the full nonlinear problem is solved only on a coarse grid of sizeH. The nonlinearities are expanded about the coarse grid solution on the fine gird of sizeh. The resulting linear system is solved on the fine grid. Some a priori error estimates are derived with theH1-normO(h+H2)for the two-grid finite element method. Compared with the standard finite element method, the two-grid method achieves asymptotically same order as long as the mesh sizes satisfyh=O(H2).

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.

A mixed finite-element method for elliptic operators with Wentzell boundary condition

2018 ◽  
Vol 40 (1) ◽  
pp. 87-108
Author(s):  
Eberhard Bänsch ◽  
Markus Gahn
Keyword(s):  
Boundary Condition ◽  
Finite Element ◽  
Mixed Finite Element ◽  
A Priori ◽  
Mixed Finite Element Method ◽  
Finite Element Approximation ◽  
Element Approximation ◽  
Weak Formulation ◽  
Wentzell Boundary Condition ◽  
Polyhedral Domain

Abstract In this paper we introduce and analyze a mixed finite-element approach for a coupled bulk-surface problem of second order with a Wentzell boundary condition. The problem is formulated on a domain with a curved smooth boundary. We introduce a mixed formulation that is equivalent to the usual weak formulation. Furthermore, optimal a priori error estimates between the exact solution and the finite-element approximation are derived. To this end, the curved domain is approximated by a polyhedral domain introducing an additional geometrical error that has to be bounded. A computational result confirms the theoretical findings.

scholarly journals AN H1 -GALERKIN MIXED FINITE ELEMENT APPROXIMATION OF A NONLOCAL HYPERBOLIC EQUATION

2017 ◽  
Vol 22 (5) ◽  
pp. 643-653
Author(s):  
Fengxin Chen ◽  
Zhaojie Zhou
Keyword(s):  
Finite Element ◽  
Hyperbolic Equation ◽  
Mixed Finite Element ◽  
A Priori ◽  
Finite Element Approximation ◽  
Nonlinear Hyperbolic Equation ◽  
Element Approximation ◽  
Discrete Scheme ◽  
Fully Discrete ◽  
Priori Error Estimates

In this paper we investigate a semi-discrete H1 -Galerkin mixed finite element approximation of one kind of nolocal second order nonlinear hyperbolic equation, which is often used to describe vibration of an elastic string. A priori error estimates for the semi-discrete scheme are derived. A fully discrete scheme is constructed and one numerical example is given to verify the theoretical findings.

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