Mixed finite element methods for quasilinear second-order elliptic problems

Milner, F. A.

doi:10.1090/S0025-5718-1985-0777266-1

Mixed finite element methods for quasilinear second-order elliptic problems
HTML articles powered by AMS MathViewer

by F. A. Milner PDF

Math. Comp. 44 (1985), 303-320 Request permission

Abstract:

A mixed finite element method is developed to approximate the solution of a quasilinear second-order elliptic partial differential equation. The existence and uniqueness of the approximation are demonstrated and optimal rate error estimates are derived.

References

S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math. 12 (1959), 623–727. MR 125307, DOI 10.1002/cpa.3160120405
F. Brezzi, On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 8 (1974), no. R-2, 129–151 (English, with French summary). MR 365287
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., $H^{1}$-Galerkin methods for a nonlinear Dirichlet problem, 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. 64–86. MR 0471369
J. Douglas Jr. and J. E. Roberts, Mixed finite element methods for second order elliptic problems, Mat. Apl. Comput. 1 (1982), no. 1, 91–103 (English, with Portuguese summary). MR 667620
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
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, Grundlehren der Mathematischen Wissenschaften, Vol. 224, Springer-Verlag, Berlin-New York, 1977. MR 0473443
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

Non Homogeneous Boundary Value Problems and Applications

J.-C. Nédélec, Mixed finite elements in $\textbf {R}^{3}$, Numer. Math. 35 (1980), no. 3, 315–341. MR 592160, DOI 10.1007/BF01396415
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
Reinhard Scholz, $L_{\infty }$-convergence of saddle-point approximations for second order problems, RAIRO Anal. Numér. 11 (1977), no. 2, 209–216, 221 (English, with French summary). MR 448942, DOI 10.1051/m2an/1977110202091
R. Scholz, A remark on the rate of convergence for a mixed finite-element method for second order problems, Numer. Funct. Anal. Optim. 4 (1981/82), no. 3, 269–277. MR 665363, DOI 10.1080/01630568208816117
R. Scholz, Optimal $L_{\infty }$-estimates for a mixed finite element method for second order elliptic and parabolic problems, Calcolo 20 (1983), no. 3, 355–377 (1984). MR 761790, DOI 10.1007/BF02576470

Sur l’Analyse Numérique des Méthodes d’Éléments Finis Hybrides et Mixtes

H. Triebel, Interpolation theory, function spaces, differential operators, VEB Deutscher Verlag der Wissenschaften, Berlin, 1978. MR 500580