A posteriori error estimates for a mixed-FEM formulation of a nonlinear
elliptic problem
Rodolfo Araya, Tomás P. Barrios, Gabriel N. Gatica, Norbert Heuer
Comput. Methods Appl. Mech. Engrg. 191 (21-22), 2317-2336, 2002.
We consider the numerical solution, via the mixed finite element method,
of a nonlinear elliptic partial differential equation in divergence form
with Dirichlet boundary conditions. Besides the temperature and
the flux, we introduce the gradient of the temperature as a further unknown,
which yields
a variational formulation with a two-fold saddle point structure. We derive a
reliable a-posteriori error estimate that depends on the solution of a local
linear boundary value problem, which does not need any equilibrium property
for its solvability. In addition, for specific finite element subspaces of
Raviart-Thomas type we are able to provide a simpler a-posteriori error
estimate that does not require the explicit solution of the local problems.
Our approach does not need the exact finite element solution, but any
reasonable approximation of it, such as, for instance, the one obtained with
a fully discrete Galerkin scheme. In particular, we suggest a scheme that
uses quadrature formulas to evaluate all the linear and semi-linear forms
involved. Finally, several numerical results illustrate the suitability of
the error estimator for the adaptive computation of the corresponding discrete
solutions.