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.

中文翻译：

---

Banach 空间中线性问题的离散化：残差最小化、非线性 Petrov--Galerkin 和单调混合方法

这项工作提出了巴拿赫空间中抽象线性算子方程的综合离散化理论。该理论的基本出发点是对偶范数中的残差最小化思想，及其使用离散对偶范数的不精确版本。结果表明，在具有严格凸对偶的严格凸自反 Banach 空间的情况下，这种发展产生了一类非线性 Petrov-Galerkin 方法，等效地，产生了具有单调非线性的抽象混合方法。制定这些方法的关键是（非线性）双射二元映射。在Fortin条件下，我们证明了抽象不精确方法的离散稳定性，并随后进行了完整的误差分析。作为我们分析的一部分，我们证明了最佳近似投影仪的新界限，其中涉及常数，取决于基础 Banach 空间的几何形状。该理论概括和扩展了经典的 Petrov-Galerkin 方法以及现有的残差最小化方法，例如不连续的 Petrov-Galerkin 方法。