Mathematics of Computation

Author(s): Carlo Lovadina; Rolf Stenberg.
Journal: Math. Comp. 75 (2006), 1659-1674.
MSC (2000): Primary 65N30
Posted: June 26, 2006
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Abstract: This paper deals with the a posteriori error analysis of mixed finite element methods for second order elliptic equations. It is shown that a reliable and efficient error estimator can be constructed using a postprocessed solution of the method. The analysis is performed in two different ways: under a saturation assumption and using a Helmholtz decomposition for vector fields.

1.: A. ALONSO, Error estimators for a mixed method, Numer. Math., 74 (1994), pp. 385-395. MR 1414415 (97g:65212)
2.: D. N. ARNOLD AND F. BREZZI, Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates, RAIRO Modél. Math. Anal. Numér., 19 (1985), pp. 7-32. MR 0813687 (87g:65126)
3.: I. BABUšKA, J. OSBORN, AND J. PITKÄRANTA, Analysis of mixed methods using mesh dependent norms, Math. Comp., 35 (1980), pp. 1039-1062. MR 0583486 (81m:65166)
4.: D. BRAESS, Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics, Cambridge University Press, 2001.MR 1827293 (2001k:65002)
5.: D. BRAESS AND R. VERFÜRTH, A posteriori error estimators for the Raviart-Thomas element, SIAM J. Numer. Anal., 33 (1996), pp. 2431-2444.MR 1427472 (97m:65201)
6.: J. H. BRAMBLE AND J. XU, A local post-processing technique for improving the accuracy in mixed finite-element approximations, SIAM J. Numer. Anal., 26 (1989), pp. 1267-1275. MR 1025087 (90m:65193)
7.: F. BREZZI, J. DOUGLAS, JR., R. DURÁN, AND M. FORTIN, Mixed finite elements for second order elliptic problems in three variables, Numer. Math., 51 (1987), pp. 237-250. MR 0890035 (88f:65190)
8.: F. BREZZI AND M. FORTIN, Mixed and Hybrid Finite Element Methods, Springer-Verlag, 1991. MR 1115205 (92d:65187)
9.: F. BREZZI, J. DOUGLAS, JR., AND L. MARINI, Two families of mixed finite elements for second order elliptic problems., Numer. Math., 47 (1985), pp. 217-235. MR 0799685 (87g:65133)
10.: C. CARSTENSEN, A posteriori error estimate for the mixed finite element method, Math. Comp., 66 (1997), pp. 465-476. MR 1408371 (98a:65162)
11.: E. DARI, R. DURÁN, C. PADRA, AND V. VAMPA, A posteriori error estimators for nonconforming finite element methods, RAIRO Modél. Math. Anal. Numér., 30 (1996), pp. 385-400. MR 1399496 (97f:65066)
12.: V. GIRAULT AND P. RAVIART, Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms, Springer-Verlag, 1986.MR 0851383 (88b:65129)
13.: C. LOVADINA AND R. STENBERG, A posteriori error analysis of the linked interpolation technique for plate bending problems, SIAM J. Numer. Anal., 43 (2005), pp. 2227-2249. MR 2192338
14.: J.-C. NÉDÉLEC, A new family of mixed finite elements in , Numer. Math., 50 (1986), pp. 57-81. MR 0864305 (88e:65145)
15.: P. RAVIART AND J. THOMAS, A mixed finite element method for second order elliptic problems, in Mathematical Aspects of the Finite Element Method. Lecture Notes in Math. 606, Springer-Verlag, 1977, pp. 292-315.MR 0483555 (58:3547)
16.: R. STENBERG, Some new families of finite elements for the Stokes equations, Numer. Math., 56 (1990), pp. 827-838. MR 1035181 (91d:65176)
17.: R. STENBERG, Postprocessing schemes for some mixed finite elements, RAIRO Modél. Math. Anal. Numér., 25 (1991), pp. 151-167.MR 1086845 (92a:65303)

Carlo Lovadina
Affiliation: Dipartimento di Matematica, Università di Pavia and IMATI-CNR, VIa Ferrata 1, Pavia 27100, Italy
Email: carlo.lovadina@unipv.it

Rolf Stenberg
Affiliation: Institute of Mathematics, Helsinki University of Technology, P.O. Box 1100, 02015 TKK, Finland
Email: rolf.stenberg@tkk.fi

DOI: 10.1090/S0025-5718-06-01872-2
PII: S 0025-5718(06)01872-2
Keywords: Mixed finite element methods, a posteriori error estimates, postprocessing.
Received by editor(s): October 20, 2004
Received by editor(s) in revised form: June 7, 2005.
Posted: June 26, 2006
Additional Notes: This work has been supported by the European Project HPRN-CT-2002-00284 ``New Materials, Adaptive Systems and their Nonlinearities. Modelling, Control and Numerical Simulation''.
Copyright of article: Copyright 2006, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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