×

Pointwise accuracy of a finite element method for nonlinear variational inequalities. (English) Zbl 0661.65065

One considers a partial differential equation equivalent to a nonlinear variational inequality -div F(\(\nabla u)+b(u)\ni f\), and a numerical approximation is proposed by combining continuous piecewise linear finite elements with a preliminary regularization of b. The result is quasi optimal in \(L^{\infty}\); and the case of locally coercive vector fields is considered.
Reviewer: G.Jumarie

MSC:

65K10 Numerical optimization and variational techniques
49J40 Variational inequalities
35R45 Partial differential inequalities and systems of partial differential inequalities
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Baiocchi, C.: Estimation d’erreur dansL ? pour les in?quations a obstacle. Mathematical Aspects of Finite Element Methods. Lect. Notes Math.606, 27-34 (1977)
[2] Brezis, H., Kinderlehrer, D.: The smoothness of solutions to nonlinear variational inequalities. Indiana Univ. Math. J.23, 831-844 (1974) · Zbl 0278.49011
[3] Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. Amsterdam: North-Holland 1978 · Zbl 0383.65058
[4] Clement, P.: Approximation by finite element functions using local regularization. RAIRO Anal. Numer.9, 33-76 (1975) · Zbl 0368.65008
[5] Dobrowolski, M., Rannacher, R.: Finite element methods for nonlinear elliptic systems of second order. Math. Nachr.94, 155-172 (1980) · Zbl 0444.65077
[6] Freshe, J., Rannacher, R.: AsymptoticL ?-error estimates for linear finite element approximations of quasilinear boundary value problems. SIAM J. Numer. Anal.15, 418-431 (1978) · Zbl 0386.65049
[7] Friedman, A.: Variational Principles and Free-Boundary Problems. New York: Wiley 1982 · Zbl 0564.49002
[8] Gerhardt, C.: Hypersurfaces of prescribed mean curvature over obstacles. Math. Z.133, 169-185 (1973) · Zbl 0265.35027
[9] Gerhardt, C.: GlobalC 1.1-regularity for solutions of quasilinear variational inequalities. Arch. Ration. Mech. Anal.89, 83-92 (1985) · Zbl 0597.49007
[10] Giaquinta, M., Pepe, L.: Esistenza e regolarit? per il problema dell’ area minima con ostacoli in n variabili. Ann. Scuola Norm. Sup. Pisa Cl. Sci.25, 481-507 (1971) · Zbl 0283.49032
[11] Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Berlin Heidelberg New York: Springer 1983 · Zbl 0562.35001
[12] Giusti, E.: Minimal Surfaces and Functions of Bounded Variation. Basel: Birkh?user 1984 · Zbl 0545.49018
[13] Glowinski, R.: Numerical Methods for Nonlinear Variational Problems. Heidelberg Berlin New York: Springer 1984 · Zbl 0536.65054
[14] Johnson, C., Thomee, V.: Error estimates for a finite element approximation of a minimal surface. Math. Comput.29, 343-349 (1975) · Zbl 0302.65086
[15] Jouron, C.: R?solution num?rique du probl?me des surfaces minima. Arch. Ration. Mech. Anal.59, 311-341 (1975) · Zbl 0353.49037
[16] Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and their Applications. New York: Academic Press 1980 · Zbl 0457.35001
[17] Nitsche, J.:L ?-convergence of finite element approximations. Mathematical aspects of finite element methods. Lect. Notes Math.606, 261-274 (1977)
[18] Nochetto, R.H.: A note on the approximation of free boundaries by finite element methods. RAIRO Model. Math. Anal. Numer.20, 355-368 (1986) · Zbl 0596.65092
[19] Nochetto, R.H.: SharpL ?-error estimates for semilinear elliptic problems with free boundaries. Numer. Math. 54: 243-255 (1988) · Zbl 0663.65125
[20] Ortega, J., Rheinboldt, W.: Iterative Solution of Nonlinear Equations in Several Variables. New York: Academic Press 1970 · Zbl 0241.65046
[21] Rannacher, R.: ZurL ?-Konvergenz linearer finiter Elemente beim Dirichlet-Problem. Math. Z.149, 69-77 (1976) · Zbl 0321.65055
[22] Rannacher, R.: Some asymptotic error estimates for finite element approximation of minimal surfaces. RAIRO Anal. Numer.11, 181-196 (1977) · Zbl 0356.35034
[23] Rannacher, R., Scott, R.: Some optimal error estimates for piecewise linear finite element approximations. Math. Comput.38, 437-445 (1982) · Zbl 0483.65007
[24] Serrin, J.: The problem of Dirichlet for quasilinear elliptic differential equations with many independent variables. Philos. Trans. Roy. Soc. London Ser. A.264, 413-496 (1969) · Zbl 0181.38003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.