The discontinuous Petrov-Galerkin method is a minimal residual method with broken test spaces and is introduced for a nonlinear mod
el problem in this paper.
Its lowest-order version applies to a nonlinear uniformly convex model example and
is equivalently characterized as a mixed formulation, a reduced formulation, and
a weighted nonlinear least-squares method. Quasi-optimal a priori and reliable and
efficient a posteriori estimates are obtained for the abstract nonlinear dPG framework
for the approximation of a regular solution. The variational model example allows
for a built-in guaranteed error control despite inexact solve. The subtle uniqueness
of discrete minimizers is monitored in numerical examples.