On the continuity in 𝐵𝑉(Ω) of the 𝐿²-projection into finite element spaces

Cockburn, Bernardo

doi:10.1090/S0025-5718-1991-1094943-9

On the continuity in ${\rm BV}(\Omega)$ of the $L\sp 2$ -projection into finite element spaces

Author: Bernardo Cockburn
Journal: Math. Comp. 57 (1991), 551-561
MSC: Primary 65N30; Secondary 46E99, 47B38, 65M60
DOI: https://doi.org/10.1090/S0025-5718-1991-1094943-9
MathSciNet review: 1094943
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We show how to obtain continuity in the ${\text{BV}}(\Omega )$ -seminorm of the ${L^2}$ -projection of $u \in {\text{BV}}(\Omega )$ into a large class of finite element spaces.

[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), 217-235. 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), 237-250. MR 890035 (88f:65190)
[3] P. Ciarlet, The finite element method for elliptic problems, North-Holland, Amsterdam, 1975. MR 0520174 (58:25001)
[4] B. Cockburn, The quasimonotone schemes for scalar conservation laws. Part I, SIAM J. Numer. Anal. 26 (1990), 1325-1341. 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), 1-21. MR 551288 (81b:65079)
[7] M. Crouzeix and V. Thomée, The stability in ${L_p}$ and of the ${L_2}$ -projection onto finite element function spaces, Math. Comp. 48 (1987), 521-532. MR 878688 (88f:41016)
[8] J. Descloux, On finite element matrices, SIAM J. Numer. Anal. 9 (1972), 260-265. MR 0309292 (46:8402)
[9] J. Douglas, Jr., T. Dupont, and L. Wahlbin, Optimal ${L_\infty }$ error estimates for Galerkin approximations to solutions of two-point boundary value problems, Math. Comp. 29 (1975), 475-483. MR 0371077 (51:7298)
[10] J. Douglas, Jr., T. Dupont, and L. Wahlbin, The stability in ${L^q}$ of the ${L^2}$ -projection into finite element function spaces, Numer. Math. 23 (1975), 193-197. MR 0383789 (52:4669)
[11] T. Dupont and R. Scott, Approximation by polynomials in Sobolev spaces, Math. Comp. 34 (1980), 441-463. 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 first-order quasi-linear equation, USSR Comput. Math. and Math. Phys. 16 (1976), 105-119.
[15] J.C. Nédélec, A new family of mixed finite elements in ${\mathbb{R}^3}$ , Numer. Math. 50 (1986), 57-81. MR 864305 (88e:65145)
[16] R. Sanders, On convergence of monotone finite difference schemes with variable spatial differencing, Math. Comp. 40 (1983), 91-106. MR 679435 (84a:65075)