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

Milner, F. A.; Park, E.-J.

doi:10.1090/S0025-5718-1995-1303087-3

A mixed finite element method for a strongly nonlinear second-order elliptic problem
HTML articles powered by AMS MathViewer

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