We study nonlinear measure data elliptic problems involving the operator of generalized Orlicz growth. Our framework embraces reflexive Orlicz spaces, as well as natural variants of variable exponent and double-phase spaces. Approximable and renormalized solutions are proven to exist and coincide for arbitrary measure datum and to be unique when for a class of data being diffuse with respect to a relevant nonstandard capacity. A capacitary characterization of diffuse measures is provided.

In this article we study discontinuous Galerkin finite element discretizations of linearsecond-order elliptic partial differential equations with Dirac delta right-hand side. Inparticular, assuming that the underlying computational mesh is quasi-uniform, we derive ana priori bound on the error measured in terms of theL2-norm. Additionally, we develop residual-based aposteriori error estimators that can be used within an adaptive mesh refinementframework. Numerical examples for the symmetric interior penalty scheme are presentedwhich confirm the theoretical results.

In this paper, we extend the reduced-basis approximations developed earlier for linear elliptic and parabolic partial differential equations with affine parameterdependence to problems involving (a) nonaffine dependence on theparameter, and (b) nonlinear dependence on the field variable.The method replaces the nonaffine and nonlinear terms with a coefficient function approximation which then permits an efficient offline-online computationaldecomposition. We first review the coefficient function approximation procedure: the essential ingredients are (i) a good collateralreduced-basis approximation space, and (ii) a stable and inexpensiveinterpolation procedure. We then apply this approach to linear nonaffine and nonlinear elliptic and parabolic equations; in eachinstance, we discuss the reduced-basis approximation and the associated offline-online computationalprocedures. Numerical results are presented to assess our approach.