# Article

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Keywords:
nonlinear boundary value problem; finite elements; rate of convergence; anisotropic heat conduction
Summary:
A nonlinear elliptic partial differential equation with homogeneous Dirichlet boundary conditions is examined. The problem describes for instance a stationary heat conduction in nonlinear inhomogeneous and anisotropic media. For finite elements of degree $k\ge 1$ we prove the optimal rates of convergence $\mathcal O(h^k)$ in the $H^1$-norm and $\mathcal O(h^{k+1})$ in the $L^2$-norm provided the true solution is sufficiently smooth. Considerations are restricted to domains with polyhedral boundaries. Numerical integration is not taken into account.
References:
[1] L. Boccardo, T. Gallouët and F. Murat: Unicité de la solution de certaines équations elliptiques non linéaires. C. R. Acad. Sci. Paris Ser. I Math. 315 (1992), 1159–1164. MR 1194509
[2] Z. Chen: On the existence, uniqueness and convergence of nonlinear mixed finite element methods. Mat. Apl. Comput. 8 (1989), 241–258. MR 1067288 | Zbl 0709.65080
[3] P. G. Ciarlet: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam, 1978. MR 0520174 | Zbl 0383.65058
[4] J. Douglas and T. Dupont: A Galerkin method for a nonlinear Dirichlet problem. Math. Comp. 29 (1975), 689–696. MR 0431747
[5] J. Douglas, T. Dupont and J. Serrin: Uniqueness and comparison theorems for nonlinear elliptic equations in divergence form. Arch. Rational Mech. Anal. 42 (1971), 157–168. MR 0393829
[6] M. Feistauer, M. Křížek and V. Sobotíková: An analysis of finite element variational crimes for a nonlinear elliptic problem of a nonmonotone type. East-West J. Numer. Math. 1 (1993), 267–285. MR 1318806
[7] M. Feistauer and V. Sobotíková: Finite element approximation of nonlinear elliptic problems with discontinuous coefficients. RAIRO Modèl. Math. Anal. Numér. 24 (1990), 457–500. MR 1070966
[8] M. Feistauer and A. Ženíšek: Compactness method in finite element theory of nonlinear elliptic problems. Numer. Math. 52 (1988), 147–163. MR 0923708
[9] J. Franců: Weakly continuous operators. Applications to differential equations. Appl. Math. 39 (1994), 45–56. MR 1254746
[10] J. Frehse and R. Rannacher: Asymptotic $L^\infty$-error estimates for linear finite element approximations of quasilinear boundary value problems. SIAM J. Numer. Anal. 15 (1978), 418–431. MR 0502037
[11] D. Gilbarg and N. S. Trudinger: Elliptic Partial Differential Equations of Second Order. Springer-Verlag, Berlin, 1977. MR 0473443
[12] I. Hlaváček: Reliable solution of a quasilinear nonpotential elliptic problem of a nonmonotone type with respect to the uncertainty in coefficients. accepted by J. Math. Anal. Appl. MR 1464890
[13] I. Hlaváček and M. Křížek: On a nonpotential and nonmonotone second order elliptic problem with mixed boundary conditions. Stability Appl. Anal. Contin. Media 3 (1993), 85–97.
[14] I. Hlaváček, M. Křížek and J. Malý: On Galerkin approximations of quasilinear nonpotential elliptic problem of a nonmonotone type. J.Math. Anal. Appl. 184 (1994), 168–189. MR 1275952
[15] M. Křížek and Q. Lin: On diagonal dominance of stiffness matrices in 3D. East-West J. Numer. Math. 3 (1995), 59–69. MR 1331484
[16] M. Křížek and L. Liu: On a comparison principle for a quasilinear elliplic boundary value problem of a nonmonotone type. Applicationes Mathematicae 24 (1996), 97–107. MR 1404987
[17] M. Křížek and P. Neittaanmäki: Mathematical and Numerical Modelling in Electrical Engineering: Theory and Applications. Kluwer, Dordrecht, 1996. MR 1431889
[18] M. Křížek and V. Preiningerová: 3d solution of temperature fields in magnetic circuits of large transformers (in Czech). Elektrotechn. obzor 76 (1987), 646–652.
[19] F. A. Milner: Mixed finite element methods for quasilinear second-order elliptic problems. Math. Comp. 44 (1985), 303–320. MR 0777266 | Zbl 0567.65079
[20] J. Nečas: Les Méthodes Directes en Théorie des Équations Elliptiques. Academia, Prague, 1967. MR 0227584
[21] J. Nečas: Introduction to the Theory of Nonlinear Elliptic Equations. Teubner, Leipzig, 1983. MR 0731261
[22] J. A. Nitsche: On $L_\infty$-convergence of finite element approximations to the solution of nonlinear boundary value problem. in: Proc. of Numer. Anal. Conf. (ed. J. H. Miller), Academic Press, New York, 1977, 317–325. MR 0513215
[23] R. H. Nochetto: Introduzione al Metodo Degli Elementi Finiti. Lecture Notes, Trento Univ., 1985.
[24] V. Preiningerová, M. Křížek and V. Kahoun: Temperature distribution in large transformer cores. Proc. of GANZ Conf. (ed. M. Franyó), Budapest, 1985, 254–261.
[25] K. Yosida: Functional Analysis. Springer-Verlag, Berlin, 1965. Zbl 0126.11504
[26] A. Ženíšek: Nonlinear Elliptic and Evolution Problems and Their Finite Element Approximations. Academic Press, London, 1990. MR 1086876
[27] A. Ženíšek: The finite element method for nonlinear elliptic equations with discontinuous coeffcients. Numer. Math. 58 (1990), 51–77.

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