scholarly journals A STABILIZING SUBGRID FOR CONVECTION–DIFFUSION PROBLEM

2006 ◽  
Vol 16 (02) ◽  
pp. 211-231 ◽  
Author(s):  
ALI I. NESLITURK
Keyword(s):  
Numerical Experiments ◽  
A Priori ◽  
Diffusion Problem ◽  
Triangular Element ◽  
Discrete Solution ◽  
Convergence Properties ◽  
Small Diffusion ◽  
Galerkin Finite Element ◽  
Convection Diffusion ◽  
Priori Error Estimates

A stabilizing subgrid which consists of a single additional node in each triangular element is analyzed by solving the convection–diffusion problem, especially in the case of small diffusion. The choice of the location of the subgrid node is based on minimizing the residual of a local problem inside each element. We study convergence properties of the method under consideration and its connection with previously suggested stabilizing subgrids. We prove that the standard Galerkin finite element solution on augmented grid produces a discrete solution that satisfy the same a priori error estimates that are typically obtained with SUPG and RFB methods. Some numerical experiments that confirm the theoretical findings are also presented.

scholarly journals A Novel Characteristic Expanded Mixed Method for Reaction-Convection-Diffusion Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Yang Liu ◽  
Hong Li ◽  
Wei Gao ◽  
Siriguleng He ◽  
Zhichao Fang
Keyword(s):  
Error Estimates ◽  
Mixed Method ◽  
Mixed Finite Element ◽  
A Priori ◽  
A Priori Error Estimates ◽  
The Novel ◽  
Convection Diffusion ◽  
Backward Euler ◽  
Priori Error Estimates ◽  
Diffusion Problems

A novel characteristic expanded mixed finite element method is proposed and analyzed for reaction-convection-diffusion problems. The diffusion term∇·(a(x,t)∇u)is discretized by the novel expanded mixed method, whose gradient belongs to the square integrable space instead of the classicalH(div;Ω)space and the hyperbolic partd(x)(∂u/∂t)+c(x,t)·∇uis handled by the characteristic method. For a priori error estimates, some important lemmas based on the novel expanded mixed projection are introduced. The fully discrete error estimates based on backward Euler scheme are obtained. Moreover, the optimal a priori error estimates inL2- andH1-norms for the scalar unknownuand a priori error estimates in(L2)2-norm for its gradientλand its fluxσ(the coefficients times the negative gradient) are derived. Finally, a numerical example is provided to verify our theoretical results.

scholarly journals COMPARISON OF ADAPTIVE MESHES FOR A SINGULARLY PERTURBED REACTION–DIFFUSION PROBLEM

2012 ◽  
Vol 17 (5) ◽  
pp. 732-748 ◽  
Author(s):  
Andrej Bugajev ◽  
Raimondas Čiegis
Keyword(s):  
Numerical Experiments ◽  
Reaction Diffusion ◽  
Adaptive Mesh ◽  
Diffusion Problem ◽  
Adaptive Meshes ◽  
Finite Element Scheme ◽  
Differential Problem ◽  
Galerkin Finite Element ◽  
Error Estimators ◽  
Shishkin Meshes

We consider a singular second-order boundary value problem. The differential problem is approximated by the Galerkin finite element scheme. The main goal is to compare the well known apriori Bakhvalov and Shishkin meshes with the adaptive mesh based on the aposteriori dual error estimators. Results of numerical experiments are presented.

scholarly journals The Sdfem for a Convection-diffusion Problem with Two Small Parameters

2003 ◽  
Vol 3 (3) ◽  
pp. 443-458 ◽  
Author(s):  
Hans-Görg Roos ◽  
Zorica Uzelac
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Numerical Experiments ◽  
Diffusion Problem ◽  
Shishkin Mesh ◽  
Singularly Perturbed ◽  
Convection Diffusion ◽  
Small Parameters ◽  
Perturbation Parameters ◽  
Theoretical Results

AbstractA singularly perturbed convection-diffusion problem with two small parameters is considered. The problem is solved using the streamline-diffusion finite element method on a Shishkin mesh. We prove that the method is convergent independently of the perturbation parameters. Numerical experiments support these theoretical results.

scholarly journals Space-Time Finite Element Methods for Parabolic Problems

2015 ◽  
Vol 15 (4) ◽  
pp. 551-566 ◽  
Author(s):  
Olaf Steinbach
Keyword(s):  
Finite Element ◽  
Numerical Experiments ◽  
Numerical Scheme ◽  
Evolution Equations ◽  
A Priori ◽  
Space Time ◽  
Parabolic Problems ◽  
Parabolic Evolution Equations ◽  
Priori Error Estimates ◽  
The Stability

AbstractWe propose and analyze a space-time finite element method for the numerical solution of parabolic evolution equations. This approach allows the use of general and unstructured space-time finite elements which do not require any tensor product structure. The stability of the numerical scheme is based on a stability condition which holds for standard finite element spaces. We also provide related a priori error estimates which are confirmed by numerical experiments.

DISCRETIZATION OF AN UNSTEADY FLOW THROUGH A POROUS SOLID MODELED BY DARCY'S EQUATIONS

2008 ◽  
Vol 18 (12) ◽  
pp. 2087-2123 ◽  
Author(s):  
CHRISTINE BERNARDI ◽  
VIVETTE GIRAULT ◽  
KUMBAKONAM R. RAJAGOPAL
Keyword(s):  
Error Estimates ◽  
A Priori ◽  
A Posteriori Error Estimates ◽  
Posteriori Error ◽  
Convergence Properties ◽  
A Posteriori Error ◽  
Flow Through ◽  
Priori Error Estimates ◽  
Dependent Flow ◽  
Darcy’S Equations

The system of unsteady Darcy's equations considered here models the time-dependent flow of an incompressible fluid such as water in a rigid porous medium. We propose a discretization of this problem that relies on a backward Euler's scheme for the time variable and finite elements for the space variables. We prove a priori error estimates that justify the optimal convergence properties of the discretization and a posteriori error estimates that allow for an efficient adaptivity strategy both for the time steps and the meshes.

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