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BDM mixed methods for a nonlinear elliptic problem. (English) Zbl 0819.65129

The author introduces a mixed formulation for a nonlinear two-dimensional elliptic Dirichlet problem. The formulation is based on a mixed finite element introduced by F. Brezzi, J. Douglas jun. and L. D. Marini [Numer. Math. 47, 217-235 (1985; Zbl 0599.65072)] but includes an auxiliary vector variable which plays the role of Lagrange multiplier. This auxiliary variable is approximated by discontinuous polynomials and so can be eliminated by static condensation with little practical increase in computational effort.
The author proves existence and uniqueness of solutions to the mixed formulation as well as optimal error estimates in \(L^ 2\), \(L^ \infty\), and \(H^{-s}\) of the approximate solutions. Two numerical examples are presented to illustrate the results.

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

Citations:

Zbl 0599.65072
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References:

[1] Brezzi, F.; Douglas, J.; Marini, L. D., Two families of mixed finite elements for second order elliptic problems, Numer. Math., 47, 217-235 (1985) · Zbl 0599.65072
[2] Chen, Z., On the existence, uniqueness and convergence of nonlinear mixed finite element methods, Mat. Apl. Comput., 8, 241-258 (1989) · Zbl 0709.65080
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[5] Douglas, J.; Dupont, T.; Serrin, J., Uniqueness and comparison theorems for nonlinear elliptic equations in divergence form, Arch. Rational Mech. Anal., 43, 157-168 (1971) · Zbl 0222.35017
[6] Douglas, J.; Roberts, J. E., Global estimates for mixed methods for second order elliptic equations, Math. Comp., 44, 39-52 (1985) · Zbl 0624.65109
[7] Gilbarg, D.; Trudinger, N., Elliptic Partial Differential Equations of Second Order, 224 (1977), Springer: Springer Berlin, Grundlehren Math. Wiss. · Zbl 0361.35003
[8] Lewy, H.; Stampacchia, G., On existence and smoothness of solutions of some noncoercive variational inequalities, Arch. Rational Mech. Anal., 41, 241-253 (1971) · Zbl 0237.49005
[9] Milner, F., Mixed finite element methods for quasilinear second-order elliptic problems, Math. Comp., 44, 303-320 (1985) · Zbl 0567.65079
[10] Morrey, C. B., Multiple Integrals in the Calculus of Variations, 130 (1966), Springer: Springer New York, Grundlehren Math. Wiss. · Zbl 0142.38701
[11] Raviart, P. A.; Thomas, J. M., A mixed finite element method for 2nd order elliptic problems, (Proc. Conf. on Mathematical Aspects of Finite Element Methods, 606 (1977), Springer: Springer Berlin), 292-315, Lecture Notes in Math. · Zbl 0362.65089
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