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

Understand

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.