Abstract In this work an explicit a posteriori error estimator for the steady incompressible Navier–Stokes equations is investigated. The error estimator is based on the variational multiscale theory, where the numerical solution is decomposed in resolved scales (FEM solution) and unresolved scales (FEM error). The error is estimated locally considering the residuals that emerge from the numerical solution and the error inverse-velocity scales, τ ’s, associated with each type of residual. These error scales are provided in this paper, which have been computed a-priori solving a set of local problems with unit residuals. Therefore, the computational effort to predict the error is small and its implementation in any FEM code is simple. As an application, a strategy to develop adaptive meshes with the aim of optimizing the computational effort is shown. Numerical examples are presented to test the behavior of the error estimator.

中文翻译：

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基于 VMS 的不可压缩 Navier-Stokes 方程的后验误差估计和自适应

摘要 在这项工作中，研究了稳定不可压缩 Navier-Stokes 方程的显式后验误差估计量。误差估计器基于变分多尺度理论，其中数值解分解为已解析尺度（FEM 解）和未解析尺度（FEM 误差）。考虑到从数值解中出现的残差和与每种残差类型相关的误差逆速度尺度 τ ，在局部估计误差。本文提供了这些误差尺度，它们是通过先验计算来解决一组具有单位残差的局部问题的。因此，预测误差的计算量很小，并且在任何 FEM 代码中的实现都很简单。作为应用程序，展示了开发自适应网格以优化计算工作的策略。给出了数值示例来测试误差估计器的行为。