## A mixed finite element method for a strongly nonlinear second-order elliptic problem

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- by F. A. Milner and E.-J. Park PDF
- Math. Comp.
**64**(1995), 973-988 Request permission

## Abstract:

The approximation of the solution of the first boundary value problem for a strongly nonlinear second-order elliptic problem in divergence form by the mixed finite element method is considered. Existence and uniqueness of the approximation are proved and optimal error estimates in ${L^2}$ are established for both the scalar and vector functions approximated by the method. Error estimates are also derived in ${L^q},2 \leq q \leq + \infty$.## References

- 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. MR**799685**, DOI 10.1007/BF01389710 - Philippe G. Ciarlet,
*The finite element method for elliptic problems*, Studies in Mathematics and its Applications, Vol. 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. MR**0520174** - 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. MR**771029**, DOI 10.1090/S0025-5718-1985-0771029-9 - Ricardo G. Durán,
*Error analysis in $L^p,\;1\leq p\leq \infty ,$ 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). MR**958875**, DOI 10.1051/m2an/1988220303711 - 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). MR**592753** - 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. MR**737190**, DOI 10.1007/978-3-642-61798-0 - 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). MR**610597** - Yonghoon Kwon and Fabio A. Milner,
*$L^\infty$-error estimates for mixed methods for semilinear second-order elliptic equations*, SIAM J. Numer. Anal.**25**(1988), no. 1, 46–53. MR**923925**, DOI 10.1137/0725005 - F. A. Milner,
*Mixed finite element methods for quasilinear second-order elliptic problems*, Math. Comp.**44**(1985), no. 170, 303–320. MR**777266**, DOI 10.1090/S0025-5718-1985-0777266-1 - Joachim Nitsche,
*$L_{\infty }$-convergence of finite element approximations*, Mathematical aspects of finite element methods (Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975) Lecture Notes in Math., Vol. 606, Springer, Berlin, 1977, pp. 261–274. MR**0488848** - 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) Lecture Notes in Math., Vol. 606, Springer, Berlin, 1977, pp. 292–315. MR**0483555**

## Additional Information

- © Copyright 1995 American Mathematical Society
- Journal: Math. Comp.
**64**(1995), 973-988 - MSC: Primary 65N30
- DOI: https://doi.org/10.1090/S0025-5718-1995-1303087-3
- MathSciNet review: 1303087