On the continuity in of the projection into finite element spaces
Author:
Bernardo Cockburn
Journal:
Math. Comp. 57 (1991), 551561
MSC:
Primary 65N30; Secondary 46E99, 47B38, 65M60
MathSciNet review:
1094943
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: We show how to obtain continuity in the seminorm of the projection of into a large class of finite element spaces.
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 [1]
 F. Brezzi, J. Douglas, Jr., and L. D. Marini, Two families of mixed finite elements for second order elliptic problems, Numer. Math. 47 (1985), 217235. MR 799685 (87g:65133)
 [2]
 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), 237250. MR 890035 (88f:65190)
 [3]
 P. Ciarlet, The finite element method for elliptic problems, NorthHolland, Amsterdam, 1975. MR 0520174 (58:25001)
 [4]
 B. Cockburn, The quasimonotone schemes for scalar conservation laws. Part I, SIAM J. Numer. Anal. 26 (1990), 13251341. MR 1025091 (91b:65106)
 [5]
 B. Cockburn, F. Coquel, and P. LeFloch, An error estimate for finite volume methods for conservation laws in several space dimensions (in preparation).
 [6]
 M. Crandall and A. Majda, Monotone difference approximations for scalar conservation laws, Math. Comp. 34 (1980), 121. MR 551288 (81b:65079)
 [7]
 M. Crouzeix and V. Thomée, The stability in and of the projection onto finite element function spaces, Math. Comp. 48 (1987), 521532. MR 878688 (88f:41016)
 [8]
 J. Descloux, On finite element matrices, SIAM J. Numer. Anal. 9 (1972), 260265. MR 0309292 (46:8402)
 [9]
 J. Douglas, Jr., T. Dupont, and L. Wahlbin, Optimal error estimates for Galerkin approximations to solutions of twopoint boundary value problems, Math. Comp. 29 (1975), 475483. MR 0371077 (51:7298)
 [10]
 J. Douglas, Jr., T. Dupont, and L. Wahlbin, The stability in of the projection into finite element function spaces, Numer. Math. 23 (1975), 193197. MR 0383789 (52:4669)
 [11]
 T. Dupont and R. Scott, Approximation by polynomials in Sobolev spaces, Math. Comp. 34 (1980), 441463. MR 559195 (81h:65014)
 [12]
 V. Girault and P. A. Raviart, Finite element methods for Navier Stokes equations, Springer Verlag Series SCM, vol. 5, 1986. MR 851383 (88b:65129)
 [13]
 E. Giusti, Minimal surfaces and functions of bounded variation, Birkhäuser, 1984. MR 775682 (87a:58041)
 [14]
 N. N. Kuznetzov, Accuracy of some approximate methods for computing the weak solutions of a firstorder quasilinear equation, USSR Comput. Math. and Math. Phys. 16 (1976), 105119.
 [15]
 J.C. Nédélec, A new family of mixed finite elements in , Numer. Math. 50 (1986), 5781. MR 864305 (88e:65145)
 [16]
 R. Sanders, On convergence of monotone finite difference schemes with variable spatial differencing, Math. Comp. 40 (1983), 91106. MR 679435 (84a:65075)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718199110949439
PII:
S 00255718(1991)10949439
Keywords:
projection,
bounded variation,
finite elements
Article copyright:
© Copyright 1991
American Mathematical Society
