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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 propose and analyze a minimal-residual method in discrete dual norms for approximating the solution of the advection-reaction equation in a weak Banach-space setting. The weak formulation allows for the direct approximation of solutions in the Lebesgue L p {L^{p}} -space, 1 < p < ∞ {1

While it is classical to consider the solution of the convection-diffusion-reaction equation in the Hilbert space H 0 1 ⁢ (Ω) {H_{0}^{1}(\Omega)} , the Banach Sobolev space W 0 1 , q ⁢ (Ω) {W^{1,q}_{0}(\Omega)} , 1 < q < ∞ {1

In this article we consider the numerical approximation of the convection-diffusion-reaction equation. One of the main challenges of designing a numerical method for this problem is that boundary layers occurring in the convection-dominated case can lead to non-physical oscillations in the numerical approximation, often referred to as Gibbs phenomena. The idea of this article is to consider the approximation problem as a residual minimization in dual norms in L q -type Sobolev spaces, with 1 < q < ∞. We then apply a non-standard, non-linear Petrov–Galerkin discretization, that is applicable to reflexive Banach spaces such that the space itself and its dual are strictly convex. Similar to discontinuous Petrov–Galerkin methods, this method is based on minimizing the residual in a dual norm. Replacing the intractable dual norm by a suitable discrete dual norm gives rise to a non-linear inexact mixed method. This generalizes the Petrov–Galerkin framework developed in the context of discontinuous Petrov–Galerkin methods to more general Banach spaces. For the convection–diffusion–reaction equation, this yields a generalization of a similar approach from the L 2 -setting to the L q -setting. A key advantage of considering a more general Banach space setting is that, in certain cases, the oscillations in the numerical approximation vanish as q tends to 1, as we will demonstrate using a few simple numerical examples. [ABSTRACT FROM AUTHOR]