BDM mixed methods for a nonlinear elliptic problem (Q1344288)

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 \textit{F. Brezzi}, \textit{J. Douglas jun.} and \textit{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.