Finite element error estimates for nonlinear elliptic equations of monotone type. (English) Zbl 0643.65058

We consider the application of the finite element method to a class of second order elliptic boundary value problems of divergence form and with gradient nonlinearity in the principal coefficient, and the derivation of error estimates for the finite element approximations. Such problems arise in many practical situations - for example, in shock-free airfoil design, seepage through coarse grained porous media, and in some glaciological problems. By making use of certain properties of the nonlinear coefficients, we demonstrate that the variational formulations associated with these boundary value problems are well-posed. We also prove that the abstract operators accompanying such problems satisfy certain continuity and monotonicity inequalities. With the aid of these inequalities and some standard results from approximation theory, we show how one may derive error estimates for the finite element approximations in the energy norm.
Reviewer: S.-S.Chow


65N15 Error bounds for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
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[1] Chaplygin, S.A.: On gas jets. Sci. Mem. Moscow Univ. Math. Phys. Sec. 21, 1902, pp. 1-121 (translation: NAC-Tech. Mem. 1063, 1944)
[2] Ciarlet, P.G.: The finite method for elliptic problems. Amsterdam: North-Holland 1978 · Zbl 0383.65058
[3] Ciarlet, P.G., Schultz, M.H., Varga, R.S.: Numerical methods of high-order accuracy for nonlinear boundary value problems: I. One dimensional problems. Numer. Math.9, 394-430 (1967) · Zbl 0155.20403
[4] Ciarlet, P.G., Schultz, M.H., Varga, R.S.: Numerical methods of high-order accuracy for nonlinear boundary value problems: V. Monotone operator theory. Numer. Math.13, 51-77 (1969) · Zbl 0181.18603
[5] Fung, K.Y., Sobieczky, H., Seebass, R.: Shock-free wing design. AIAAJ.18, 1153-1158 (1980) · Zbl 0455.76062
[6] Chow, S.-S.: Finite element error estimates for nonlinear elliptic problems of monotone type, Ph.D. Thesis. Anst. Nat. Univ., Canberra, Australia, 1983 · Zbl 0514.65079
[7] Carey, G.F., Pan, T.T.: Shock-free redesign using finite elements, Comm. Appl. Numer. Meth.2, 29-35 (1986) · Zbl 0591.76103
[8] Glowinski, R., Marrocco, A.: Sur l’approximation par ?l?ments finis d’ordre un, et la r?solution, par p?nalisation-dualit?, d’une classe de probl?mes de Dirichlet non-lin?aires. RAIRO, R2, 41-76 (1975)
[9] Johnson, C., Thom?e, V.: Error estimates for a finite element approximation of a minimal surface. Math. Comput.29, 343-349 (1975)
[10] Noor, M.A., Whiteman, J.R.: Error bounds for finite element solutions of mildly nonlinear elliptic boundary value problems. Numer. Math.26, 107-116 (1976) · Zbl 0312.65079
[11] Noor, M.A.: Finite element approximation theory for strongly nonlinear problems. Comment. Math. Univ. St. Paul31, 1-7 (1982) · Zbl 0482.65062
[12] Rockafellar, R.T.: Convex Analysis. Princeton, NJ: Princeton University Press 1970 · Zbl 0193.18401
[13] Strang, G., Fix, G.: An analysis of the finite element method. Englewood Cliffs, NJ: Prentice-Hall 1973 · Zbl 0356.65096
[14] Tyukhtin, V.B.: The rate of convergence of approximation methods of solution of one-sided variational problems I. Vestnik Leningrad Univ. Mat. Mekh. Astronom., No. 13, July 1982, pp. 111-113 (in Russian) · Zbl 0492.49016
[15] Von K?rm?n, T.: Compressibility effects in aerodynamics. J. Aero. Sci.8, 337-356 (1941) · JFM 67.0853.01
[16] Xie, G.: An analysis of finite element method for the minimization of convex functionals. Acta Math. Appl. Sinica3, 57-71 (1981) (in Chinese)
[17] Glowinski, R.: Numerische methods for nonlinear variational problems. New York Heidelberg Berlin: Springer 1984 · Zbl 0536.65054
[18] Bristeau, M.O., Glowinski, R., Perriaux, J., Pironneau, O., Poirier, G.: On the numerical solution of nonlinear problems in fluid dynamics by the least squares and finite element methods (II). Applications to transonic flow simulations. Comput. Methods in Appl. Mechn. Eng.51, 363-394 (1985) · Zbl 0555.76046
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