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.
翻译:在Bank and Weiser[Math. Comp. 44 (1985), pp.283-301] 的创举文件中,引入了一个新的后继测算器。这个测算器需要解决有限元素网格中每个细胞的局部内纽曼问题。尽管Bank-Weiser 类型测算器的允诺,即地点、计算效率和无症状锐度,但是它们在实际计算问题中几乎没有看到什么用处。这一贡献的重点是描述在现代具有自动代码生成能力的高端有限元素软件中,对银行-Weiser 类型测算器的分级测算器的新应用。我们展示了如何使用该测算器驱动(面向目标的)适应性网格的改进,以及使几乎无法压缩的弹性问题的混合近距离。我们提供了与其他使用的其他估算器的比较。根据FENICS项目有限元素软件提供的开放源实施软件作为补充材料。