×

zbMATH — the first resource for mathematics

A mixed finite element method for a strongly nonlinear second-order elliptic problem. (English) Zbl 0829.65128
The authors consider the approximation of the solution of strongly nonlinear two-dimensional second-order elliptic boundary value problems in divergence form by using the mixed finite element method. The spatial domain is considered to be a bounded, convex domain with \(C^2\)- boundary. Furthermore, the coefficient vector is assumed to have a bounded positive definite Jacobian with respect to the second argument. This assumption implies that the gradient of the solution can be locally represented as a function of the “flux”. The authors assume that such a representation is global.
Theorems for the existence and uniqueness of the approximation are proved making use of the Raviart-Thomas space of index \(k > 0\) and introducing \(L^2\) and Raviart-Thomas projections. After that, the authors extend some previous results of F. A. Milner [Math. Comput. 44, 303-320 (1985; Zbl 0567.65079)] and derive error estimates in \(L^q\), \(2 \leq q \leq \infty\) using some generalizations of the \(L^2\) lemmas (proved in the previous paragraph) and Nitsche’s weighted \(L^2\)-norms.
Reviewer: K.Georgiev (Sofia)

MSC:
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Franco Brezzi, Jim Douglas Jr., and L. D. Marini, Two families of mixed finite elements for second order elliptic problems, Numer. Math. 47 (1985), no. 2, 217 – 235. · Zbl 0599.65072 · doi:10.1007/BF01389710 · doi.org
[2] Philippe G. Ciarlet, The finite element method for elliptic problems, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. Studies in Mathematics and its Applications, Vol. 4. · Zbl 0383.65058
[3] Jim Douglas Jr. and Jean E. Roberts, Global estimates for mixed methods for second order elliptic equations, Math. Comp. 44 (1985), no. 169, 39 – 52. · Zbl 0624.65109
[4] Ricardo G. Durán, Error analysis in \?^\?,1\le \?\le \infty , for mixed finite element methods for linear and quasi-linear elliptic problems, RAIRO Modél. Math. Anal. Numér. 22 (1988), no. 3, 371 – 387 (English, with French summary). · Zbl 0698.65060
[5] R. S. Falk and J. E. Osborn, Error estimates for mixed methods, RAIRO Anal. Numér. 14 (1980), no. 3, 249 – 277 (English, with French summary). · Zbl 0467.65062
[6] David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. · Zbl 0562.35001
[7] Claes Johnson and Vidar Thomée, Error estimates for some mixed finite element methods for parabolic type problems, RAIRO Anal. Numér. 15 (1981), no. 1, 41 – 78 (English, with French summary). · Zbl 0476.65074
[8] Yonghoon Kwon and Fabio A. Milner, \?^\infty -error estimates for mixed methods for semilinear second-order elliptic equations, SIAM J. Numer. Anal. 25 (1988), no. 1, 46 – 53. · Zbl 0643.65057 · doi:10.1137/0725005 · doi.org
[9] F. A. Milner, Mixed finite element methods for quasilinear second-order elliptic problems, Math. Comp. 44 (1985), no. 170, 303 – 320. · Zbl 0567.65079
[10] Joachim Nitsche, \?_\infty -convergence of finite element approximations, Mathematical aspects of finite element methods (Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975) Springer, Berlin, 1977, pp. 261 – 274. Lecture Notes in Math., Vol. 606.
[11] P.-A. Raviart and J. M. Thomas, A mixed finite element method for 2nd order elliptic problems, Mathematical aspects of finite element methods (Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975) Springer, Berlin, 1977, pp. 292 – 315. Lecture Notes in Math., Vol. 606.
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.