椭圆问题的各种边界条件及其近似的统一分析
A unified analysis of elliptic problems with various boundary conditions and their approximation
AI总结：
本文提出一种抽象框架，用于Banach空间中对偶算子的近似分析，涵盖线性扩散等Leray-Lions型问题，统一分析各种边界条件和近似方法的收敛性与误差估计。
摘要：
我们设计了一个抽象框架，用于Banach空间中对偶算子的近似分析。典型例子包括Lebesgue-Sobolev空间上的梯度和散度算子。我们将此抽象框架应用于Leray-Lions型问题的数值近似，包括线性扩散。抽象框架的主要目的是提供一种统一的收敛分析，同时涵盖所有常规边界条件和多种近似方法。考虑的近似可以是符合的或非符合的（即近似函数可以属于问题的能量空间或不）。收敛结果和误差估计被给出。最后简要展示该抽象框架如何应用于其他模型，包括裂隙介质中的流体流动、弹性方程和流形上的扩散方程。分析的副产品是关于一般Poincaré不等式与适当空间中散度算子满射性的等价性的一个新结果。
英文摘要：
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, or not (that is, the approximation functions can belong to the energy space of 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 other models, including flows in fractured medium, elasticity equations and diffusion equations on manifolds. A by-product of the analysis is an apparently novel result on the equivalence between general Poincar{é} inequalities and the surjectivity of the divergence operator in appropriate spaces.
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