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.


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


Zbl 0599.65072
Full Text: DOI


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