Discretization of Linear Problems in Banach Spaces: Residual Minimization, Nonlinear Petrov-Galerkin, and Monotone Mixed Methods

TL;DR

This paper develops a comprehensive discretization framework for linear operator equations in Banach spaces, introducing residual minimization, nonlinear Petrov-Galerkin, and mixed methods with stability and error analysis.

Contribution

It generalizes classical Petrov-Galerkin and residual-minimization methods to Banach spaces, incorporating nonlinear duality maps and establishing stability and error bounds.

Findings

Proves discrete stability under Fortin condition.

Provides new bounds for best-approximation projectors.

Extends residual-minimization approaches like discontinuous Petrov-Galerkin.

Abstract

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. Crucial in the formulation of these methods is the (nonlinear) bijective duality map. Under the Fortin condition, we prove discrete stability of the abstract inexact method, and subsequently carry out a complete error analysis. As part of our analysis, we prove new bounds for best-approximation projectors, which involve constants depending on the geometry of the underlying Banach space. The theory generalizes and extends the classical Petrov-Galerkin method as well as existing residual-minimization approaches, such as the discontinuous Petrov-Galerkin method.