We design an abstract setting for the approximation in Banach spaces of operators acting in duality. A typical example are the gradient and divergence operators in Lebesgue-Sobolev spaces on a bounded domain. We apply this abstract setting to the numerical approximation of Leray-Lions type problems, which include in particular linear diffusion. The main interest of the abstract setting is to provide a unified convergence analysis that simultaneously covers (i) all usual boundary conditions, (ii) several approximation methods. The considered approximations can be conforming (that is, the approximation functions can belong to the energy space relative to the problem) or not, and include classical as well as recent numerical schemes. Convergence results and error estimates are given. We finally briefly show how the abstract setting can also be applied to some models such as flows in fractured medium, elasticity equations and diffusion equations on manifolds.

中文翻译：

---

各种边界条件椭圆问题的统一分析及其近似

我们设计了一个抽象设置，用于对偶操作的操作符在 Banach 空间中的近似。一个典型的例子是有界域上 Lebesgue-Sobolev 空间中的梯度和散度算子。我们将此抽象设置应用于 Leray-Lions 类型问题的数值近似，其中特别包括线性扩散。抽象设置的主要兴趣是提供统一的收敛分析，同时涵盖 (i) 所有常用边界条件，(ii) 几种近似方法。所考虑的近似值可以是符合的（即，近似函数可以属于与问题相关的能量空间），也可以不符合，并且包括经典和最近的数值方案。给出了收敛结果和误差估计。