  In this paper we develop adaptive numerical schemes for certain nonlinear
variational problems. The discretization of the variational problems is done by
representing the solution as a suitable frame decomposition, i.e., a complete,
stable, and redundant expansion. The discretization yields an equivalent
nonlinear problem on $\ell_2(\mathcal{N})$, the space of frame coefficients.
The discrete problem is then adaptively solved using approximated nested fixed
point and Richardson type iterations. We investigate the convergence,
stability, and optimal complexity of the scheme. This constitutes a theoretical
advantage, for example, with respect to adaptive finite element schemes for
which convergence and complexity results are still hard to prove. The use of
frames is further motivated by their redundancy, which, at least numerically,
has been shown to improve the conditioning of the corresponding discretization
matrices. Frames also are usually easier to construct than Riesz bases.
Finally, we show how to apply the adaptive scheme we propose for finding an
approximation to the solution of the PDEs governing magnetohydrodynamic (MHD)
flows, once suitable frame expansions are constructed.
