In the seminal paper of Bank and Weiser [Math. Comp., 44 (1985), pp.283-301] a new a posteriori estimator was introduced. This estimator requires the solution of a local Neumann problem on every cell of the finite element mesh. Despite the promise of Bank-Weiser type estimators, namely locality, computational efficiency, and asymptotic sharpness, they have seen little use in practical computational problems. The focus of this contribution is to describe a novel implementation of hierarchical estimators of the Bank-Weiser type in a modern high-level finite element software with automatic code generation capabilities. We show how to use the estimator to drive (goal-oriented) adaptive mesh refinement and to mixed approximations of the nearly-incompressible elasticity problems. We provide comparisons with various other used estimators. An open-source implementation based on the FEniCS Project finite element software is provided as supplementary material.

中文翻译：

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FEniCS项目中Bank-Weiser类型的分层后验误差估计

在Bank and Weiser的开创性论文[Math。Comp。，44（1985），pp.283-301]引入了一种新的后验估计器。该估计器需要求解有限元网格的每个像元上的局部Neumann问题。尽管有Bank-Weiser类型估计器的希望，即局部性，计算效率和渐近清晰度，但在实际的计算问题中很少使用它们。该贡献的重点是描述具有自动代码生成功能的现代高级有限元软件中Bank-Weiser类型的分层估计器的新颖实现。我们展示了如何使用估算器来驱动（面向目标的）自适应网格细化，以及如何近似压缩不可压缩的弹性问题的混合近似。我们提供与其他各种使用过的估算器的比较。