A DPG FRAMEWORK FOR STRONGLY MONOTONE OPERATORS

Pierre Cantin (1) , Norbert Heuer (1)

1 UC - Pontificia Universidad Católica de Chile

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.

Mots clés: Discontinuous Petrov–Galerkin method; optimal test functions; strongly monotone operator; advection--diffusion; nonlinear penalty

Domaines: Equations aux dérivées partielles [math.AP]; Analyse numérique [math.NA]; Analyse fonctionnelle [math.FA]

Pierre Cantin, Norbert Heuer. A DPG FRAMEWORK FOR STRONGLY MONOTONE OPERATORS. 2018. ⟨hal-01690281⟩