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This work presents a comprehensive discretization theory for abstract linear operator equations in Banach spaces. The fundamental starting point of the theory is the idea of residual minimization in dual norms and its inexact version using discrete dual norms. It is shown that this development, in the case of strictly convex reflexive Banach spaces with strictly convex dual, gives rise to a class of nonlinear Petrov-Galerkin methods and, equivalently, abstract mixed methods with monotone nonlinearity. Under the Fortin condition, we prove discrete stability and quasi-optimal convergence of the abstract inexact method, with constants depending on the geometry of the underlying Banach spaces. The theory generalizes and extends the classical Petrov-Galerkin method as well as existing residual-minimization approaches, such as the discontinuous Petrov-Galerkin method. [ABSTRACT FROM AUTHOR]

We provide an abstract framework for optimal goal-oriented adaptivity for finite element methods and boundary element methods in the spirit of [C. Carstensen et al., Comput. Math. Appl., 67 (2014), pp. 1195-1253]. We prove that this framework covers standard discretizations of general second-order linear elliptic PDEs and hence generalizes available results [R. Becker, E. Estecahandy, and D. Trujillo, SIAM J. Numer. Anal., 49 (2011), pp. 2451-2469, M. S. Mommer and R. Stevenson, SIAM J. Numer. Anal., 47 (2009), pp. 861-886] beyond the Poisson equation. [ABSTRACT FROM AUTHOR]