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Anal."],"published-print":{"date-parts":[[2018,1]]},"abstract":"<jats:p>We present and analyze a hybrid technique for numerically solving strongly monotone nonlinear problems by the discontinuous Petrov--Galerkin method with optimal test functions (DPG method). Our strategy is to relax the nonlinear problem to a linear one with additional unknowns and to consider the nonlinear relation as a constraint. We propose using optimal test functions only for the linear problem and enforcing the nonlinear constraint by penalization. In fact, our scheme can be seen as a minimum residual method with a 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. 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