A DPG framework for strongly monotone operators
Pierre Cantin and Norbert Heuer
SIAM J. Numer. Anal. 56 (5), 2731-2750, 2018.
We present and analyze a hybrid technique to numerically solve strongly monotone nonlinear problems
by the discontinuous Petrov-Galerkin method with optimal test functions (DPG).
Our strategy is to relax the nonlinear problem to a linear one with additional unknown and
to consider the nonlinear relation as a constraint. We propose to use
optimal test functions only for the linear problem and to enforce the nonlinear constraint by penalization.
In fact, our scheme can be seen as a minimum residual method
with nonlinear penalty term. We develop an abstract framework of the relaxed DPG scheme
and prove under appropriate assumptions the well-posedness of the continuous formulation
and the quasi-optimal convergence of its discretization.
As an application we consider an advection-diffusion problem with nonlinear diffusion
of strongly monotone type. Some numerical results in the lowest-order setting are presented
to illustrate the predicted convergence.