A posteriori analysis and adaptive error control for multiscale operator decomposition solution of elliptic systems I: Triangular systems

SIAM Journal on Numerical Analysis

Volumn 47, Issue 1, 2008, Pages 740-761

  Carey, V ^a   Estep, D ^a   Tavener, S ^a  

^a Veterinary Dentistry and Oral Surgery   (United States)

Author keywords

A posteriori error analysis; Adjoint problem; Elliptic system; Generalized green's function; Goal oriented error estimates; Multiscale methods; Operator decomposition; Projection error

Indexed keywords

ADJOINT PROBLEMS; ELLIPTIC SYSTEM; GENERALIZED GREEN'S FUNCTION; GOAL ORIENTED ERROR ESTIMATES; MULTISCALE METHOD; OPERATOR DECOMPOSITION; POSTERIORI ERROR ANALYSIS; PROJECTION ERROR;

ADAPTIVE CONTROL SYSTEMS; ERROR ANALYSIS; FUZZY CONTROL; GEOMETRY; GREEN'S FUNCTION;

FINITE ELEMENT METHOD;

EID: 74949094151     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/070689917     Document Type: Article

References (16)

1. W. BANGERTH AND R. RANNACHER, Adaptive Finite Element Methods for Differential Equations, Birkhauser-Verlag, New York, 2003.
2. R. BECKER AND R. RANNACHER, An optimal control approach to a posteriori error estimation in finite element methods, Acta Numer., 10 (2001), pp. 1-102.
3. V. CAREY, D. ESTEP, AND S. TAVENER, A posteriori analysis and adaptive error control for operator decomposition methods for elliptic systems II: Fully coupled systems, Internat. J. Numer. Methods Engrg., submitted.
4. V. CAREY, D. ESTEP, AND S. TAVENER, Averaging based projections in operator decomposition methods for elliptic systems, 2008, manuscript.
5. K. ERIKSSON, D. ESTEP, P. HANSBO, AND C. JOHNSON, Introduction to adaptive methods for differential equations, Acta Numer., 4 (1995), pp. 105-158.
6. K. ERIKSSON, D. ESTEP, P. HANSBO, AND C. JOHNSON, Computational Differential Equations, Cambridge University Press, Cambridge, 1996.
7. D. ESTEP, A posteriori error bounds and global error control for approximation of ordinary differential equations, SIAM J. Numer. Anal., 32 (1995), pp. 1-48.
8. D. ESTEP, M. HOLST, AND M. LARSON, Generalized Green's functions and the effective domain of influence, SIAM J. Sci. Comput., 26 (2005), pp. 1314-1339.
9. D. ESTEP, M. G. LARSON, AND R. D. WILLIAMS, Estimating the error of numerical solutions of systems of reaction-diffusion equations, Mem. Amer. Math. Soc., 146 (2000), pp. viii+109.
10. D. ESTEP, S. TAVENER, AND T. WILDEY, A posteriori analysis and improved accuracy for an operator decomposition solution of a conjugate heat transfer problem, SIAM J. Numer. Anal., 46 (2008), pp. 2068-2089.
11. G. S. FISHMAN, Monte Carlo: Concepts, algorithms, and applications, Springer Ser. Oper. Res., Springer-Verlag, New York, 1996. Concepts, algorithms, and applications.
12. M. GILES AND E. SÜLI, Adjoint methods for PDEs: A posteriori error analysis and postprocessing by duality, Acta Numer., 11 (2002), pp. 145-236.
13. V. GINTING, D. ESTEP, J. SHADID, and S. TAVENER, An a posteriori analysis of operator splitting, SIAM J. Numer. Anal., 46 (2008), pp. 1116-1146.
14. G.I. MARCHUK, On the theory of the splitting-up method, in Proceedings of the Second Symposium on Numerical Solution of Partial Differential Equations, SVNSPADE, Academic Press, New York, 1970, pp. 469-500.
15. G.I. MARCHUK, Splitting and alternating direction methods, in Handbook of Numerical Analysis, Vol.I, P. G. Ciarlet and J. L. Lions, eds., North-Holland, New York, 1990, pp. 197-462.
16. G. STRANG AND G. J. FIX, An Analysis of the Finite Element Method, Prentice-Hall Series in Automatic Computation, Prentice-Hall, Englewood Cliffs, NJ, 1973.