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International Journal of Technology Enhancements and Emerging Engineering Research (ISSN 2347-4289)
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International Journal of Technology Enhancements and Emerging Engineering Research  
International Journal of Technology Enhancements and Emerging Engineering Research

Website: http://www.ijteee.org

ISSN 2347-4289



Neumann-Type A Posteriori Error Estimation For Steady Convection-Diffusion Equation

[Full Text]

 

AUTHOR(S)

G. Temesgen Mekuria, J. Anand Rao

 

KEYWORDS

Keywords: Convection-diffusion equation, GK Method, SDFEM, a Posteriori error estimator, Effectivity Indices

 

ABSTRACT

ABSTRACT: We consider a steady linear convection – diffusion equation in 2D, present the standard Galerkin (GK) approximation and the Streamline-Diffusion Finite Element Method (SDFEM) and give an analysis of a posteriori error estimator based on solving a local Neumann problem. The estimator gives global upper and local lower bounds on the error measured in the H^1 semi-norm. Our numerical results from GK and SD approximations show that the global effectivity indices deteriorate in rates O(〖Pe〗_K) and O(√(〖Pe〗_K )) as 〖Pe〗_K→∞, respectively i.e., the estimator is over-estimated the error locally within a boundary layer which is not resolved by uniform grid refinement.

 

REFERENCES

M. Ainsworth and J. Oden, “A procedure for a posteriori error estimation for h – p finite element methods”. Comp. Meth. Eng. 101, 73-96, 1992.

M. Ainsworth and I. Babuˇska, “Reliable and roubst a posteriori error estimation for singular perturbed reaction-diffusion problems”. SIAM J. Numer. Anal., 36:331–353, 1999.

Babuˇska and A. K. Aziz, “On the angle condition in the finite element method”. SIAM. Numer. Anul., 13:214-226, 1976.

Babuˇska, R. Dur´an, and R. Rodr´ıguez, “Analysis of the efficiency of an a posteriori error estimator for linear triangular finite elements”. SIAM J. Numer. Anal., 29:947–964, 1992.

R. E. Bank and A. Weiser, “Some a posteriori error estimators for elliptic partial differential equations”, Math. Comp., 44, pp. 283–301, 1985.

P. Cl´ement, “Approximation by finite element functions using local regularization”, R.A.I.R.O. Anal. Num´er., 2 , pp. 77–84, 1975.

R. Dur´an, M. A. Muschietti, and R. Rodr´ıguez, “On the asymptotic exactness of error estimators for linear triangular finite elements”. Numer. Math., 59:107–127, 1991.

R. Dur´an and R. Rodr´ıguez, “On the asymptotic exactness of Bank-Weiser’s estimator”. Numer. Math., 62:297–303, 1992.

W. Eckhaus, “Asymptotic Analysis of Singular Perturbations”, North-Holland, Amsterdam. 1979.

H. Elman, D. Silvester, and A. Wathen, “Finite Elements and Fast Iterative Solvers: with Applications in Incompressible Fluid Dynamics”. Numerical Methods and Scientific Computation. Oxford University Press, Oxford. 2005.

K. Eriksson and C. Johnson, “Adaptive streamline diffusion finite element methods for stationary convection-diffusion problems”. Math. Comp., 60:167–188, 1993.

K. Eriksson, D. Estep, P. Hansbo, and C. Johnson, “Computational Differential Equations”, Cambridge University Press, New York. 1996.

P. M. Gresho & R. L. Sani, “ Flow and the Finite Element Method”. Jhon Wiley, Chichester. 1998.

T. J. R. Hughes and A.N. Brooks, “A multidimensional upwind scheme with no crosswind diffusion”, in Hughes T. J. R, ed., Finite element methods for convection dominated flows, AMD – vol. 34, ASME, New York, pp, 19-35, 1979.

T.J.R. Hughes, M. Mallet, and A. Mizukami, “A new finite element formulation for computational fluid dynamics: II. Beyond SUPG”. Còmput. Meths. Appi. Mech. Engrg., vol. 54, 341-355, 1986.

V. John and P. Knobloch, “On spurious oscillations at layers diminishing (SOLD) methods for convection–diffusion equations: Part I – A review”, Comput. Methods Appl. Mech. Engrg. 196: 2197–2215, 2007.

V. John and P. Knobloch, “On spurious oscillations at layers diminishing (SOLD) methods for convection–diffusion equations: Part II-analysis for P_1 and Q_1 finite elements”. Comput. Methods Appl. Mech. Engrg., 197:1997-2014, 2008.

C. Johnson, U. Nävert, and J. Pitkäranta, “Finite element methods for linear hyperbolic equations”. Còmput. Meths. Appi. Mech. Engrg., vol. 45, 285-312, 1984.

C. Johnson, “Numerical Solution of Partial Differential Equations by the Finite Element Method”. Cambridge University Press, New York. 1987.

C. Johnson, A. H. Schatz, and L. B. Wahlbin, “Crosswind smear and pointwise errors in streamline diffusion finite element methods”. Math. Comput., 49:25–38, 1987.

C. Johnson, “The streamline diffusion finite element method for compressible and incompressible fluid flow”. Finite elements in fluids, 8:75–95, 1989.

C. Johnson, “Adaptive Finite element methods for diffusion and convection problems”, Comput, Meth. Appl. Mech. Engrg. 82, 301-322, 1990.

D. Kay and D. Silvester, “A posteriori error estimation for stabilized mixed approximations of the Stokes equations”, SIAM J. Sci. Comput., 21, pp. 1321–1336, 2000.

D. Kay and D. Silvester, “The reliability of local error estimators for convection–diffusion equations”, IMA J. Numer. Anal., 21, pp. 107–122, 2001.

P. Knobloch, “On the choice of the supg parameter at outflow boundary layers”. Technical Report MATH-knm-2007/3, Charles University, Faculty of Mathematics and Physics, Prague. 2007.

H.-G. Roos, M. Stynes, and L. Tobiska, “Numerical Methods for Singularly Perturbed Differential Equations”, Springer-Verlag, Berlin. 1996.

R. Verf¨urth, “A posteriori error estimation and adaptive mesh-refinement techniques”. J. Comput. Appl. Math., 50:67–83, 1994.

R. Verf¨urth, “A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques”, Wiley–Teubner, Chichester. 1996.

R. Verf¨urth, “A posteriori error estimators for convection– diffusion equations”, Numer. Math.,80, pp.641–663, 1998.