zbMATH — the first resource for mathematics

The finite element method for nonlinear elliptic equations with discontinuous coefficients. (English) Zbl 0709.65081
The study of the finite element approximation to nonlinear second order elliptic boundary value problems with discontinuous coefficients is presented in the case of mixed Dirichlet-Neumann boundary conditions. The change in domain and numerical integration are taken into account. With the assumptions which guarantee that the corresponding operator is strongly monotone and Lipschitz-continuous the following convergence results are proved: 1. The rate of convergence \(O(h^{\epsilon})\) if the exact solution \(u\in H^ 1(\Omega)\) is piecewise of class \(H^{1+\epsilon}\) \((0<\epsilon \leq 1)\); 2. The convergence without any rate of convergence if \(u\in H^ 1(\Omega)\) only.
Reviewer: Jiping Lu

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
Full Text: DOI EuDML
[1] Babuška, I.: The finite element method for elliptic equations with discontinuous coefficients. Computing5, 207–213 (1970) · Zbl 0199.50603 · doi:10.1007/BF02248021
[2] Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. Amsterdam: North-Holland (1978) · Zbl 0383.65058
[3] Doktor, P.: On the density of smooth functions in certain subspaces of Sobolev space. Commentationes Mathematicae Universitatis Carolinae14, 609–622 (1973) · Zbl 0268.46036
[4] Feistauer, M.: On the finite element approximation of functions with noninteger derivatives. Numer. Funct. Anal. Optimization10, 91–110 (1989) · Zbl 0668.65008 · doi:10.1080/01630568908816293
[5] Feistauer, M., Sobotíková, V.: Finite element approximation of nonlinear elliptic problems with discontinuous coefficients. Preprint, Charles Univ., Prague 1988 (to appear in M2AN)
[6] Feistauer, M., Ženíšek, A.: Finite element solution of nonlinear elliptic problems. Numer. Math.50, 451–475 (1987) · Zbl 0637.65107 · doi:10.1007/BF01396664
[7] Feistauer, M., Ženíšek, A.: Compactness method in the finite element theory of nonlinear elliptic problems. Numer. Math.52, 147–163 (1988) · Zbl 0642.65075 · doi:10.1007/BF01398687
[8] Glowinski, R., Marrocco, A.: Analyse numérique du champ magnétique d’un alternateur par elements finis et surrelaxation punctuelle non lineaire. Comput. Methods Appl. Mech. Eng.3, 55–85 (1974). · Zbl 0288.65068 · doi:10.1016/0045-7825(74)90042-5
[9] Glowinski, R., Marrocco, A.: Numerical solution of two-dimensional magnetostatic problems by augmented Lagrangian methods. Comput. Methods Appl. Mech. Eng.12, 33–46 (1977) · Zbl 0364.65104 · doi:10.1016/0045-7825(77)90049-4
[10] Jamet, P.: Estimations d’erreur pour des éléments finis droits presque dégénérés. RAIRO Anal. Numér.10, 43–61 (1976)
[11] Kreisinger, V., Adam, J.: Magnetic fields in nonlinear anisotropic ferromagnetics. Acta Technica ČSAV, 209–241 (1984) · Zbl 0535.73084
[12] Kufner, A., John, O., Fučik, S.: Function Spaces. Prague: Academia 1977
[13] Marrocco, A.: Analyse numérique de problémes d’électrotechnique. Ann. Sci. Math. Québ.1, 271–296 (1977) · Zbl 0434.65093
[14] Melkes, F., Chrobáček, K., Rak L.: A stationary magnetic field computation in electrical machines. Technika elektrických strojú, VÚES Brno1–2, 22–29 (1983) (in Czech.)
[15] Nečas, J.: Les méthodes directes en théorie des equations elliptiques. Academia Prague, Masson Paris 1967
[16] Oganesian, L.A., Ruhovec, L.A.: Variational-difference methods for the solution of elliptic problems. Jerevan: Izd. Akad. Nauk ArSSR 1979 (in Russian)
[17] Rektorys, K.: Variational methods in mathematics, science and engineering, 2nd ed. Dordrecht-Boston: Reidel 1979 · Zbl 0401.35046
[18] Ženíšek, A.: Discretes forms of Friedrichs’ inequalities in the finite element method. RAIRO Numer. Anal.15, 265–286 (1981)
[19] Ženíšek, A.: How to avoid the use of Green’s theorem in the Ciarlet-Raviart theory of variational crimes. M2AN21, 171–191 (1987) · Zbl 0623.65072
[20] Ženíšek, A.: Finite element variational crimes in parabolic-elliptic problems. Part I. Nonlinear schemes. Numer. Math.55, 343–376 (1989) · Zbl 0673.65071
[21] Zlámal, M.: Curved elements in the finite element method. I. SIAM J. Numer. Anal.10, 229–240 (1973) · Zbl 0285.65067 · doi:10.1137/0710022
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.