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首页> 外文期刊>Journal of Scientific Computing >Guaranteed and Fully Robust a posteriori Error Estimates for Conforming Discretizations of Diffusion Problems with Discontinuous Coefficients
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Guaranteed and Fully Robust a posteriori Error Estimates for Conforming Discretizations of Diffusion Problems with Discontinuous Coefficients

机译：具有不连续系数扩散问题的离散化的有保证和完全鲁棒的后验误差估计

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摘要

We study in this paper a posteriori error estimates for H'~1 -conforming numerical approximations of diffusion problems with a diffusion coefficient piecewise constant on the mesh cells but arbitrarily discontinuous across the interfaces between the cells. Our estimates give a global upper bound on the error measured either as the energy norm of the difference between the exact and approximate solutions, or as a dual norm of the residual. They are guaranteed, meaning that they feature no undetermined constants. (Local) lower bounds for the error are also derived. Herein, only generic constants independent of the diffusion coefficient appear, whence our estimates are fully robust with respect to the jumps in the diffusion coefficient. In particular, no condition on the diffusion coefficient like its monotonous increasing along paths around mesh vertices is imposed, whence the present results also include the cases with singular solutions. For the energy error setting, the key requirement turns out to be that the diffusion coefficient is piecewise constant on dual cells associated with the vertices of an original simplicial mesh and that harmonic averaging is used in the scheme. This is the usual case, e.g., for the cell-centered finite volume method, included in our analysis as well as the vertex-centered finite volume, finite difference, and continuous piecewise affine finite element ones. For the dual norm setting, no such a requirement is necessary. Our estimates are based on H(div)-conforming flux reconstruction obtained thanks to the local conservativity of all the studied methods on the dual grids, which we recall in the paper; mutual relations between the different methods are also recalled. Numerical experiments are presented in confirmation of the guaranteed upper bound, full robustness, and excellent efficiency of the derived estimators.
机译：我们在本文中研究了H'〜1-扩散问题的数值近似的后验误差估计，其扩散系数在网格单元上为分段常数，但在单元之间的界面上任意不连续。我们的估计给出了误差的全局上限，可以作为精确解和近似解之间差的能量范数，也可以作为残差的对偶范数来度量。它们是有保证的，这意味着它们不具有不确定的常数。还导出错误的（本地）下限。在此，仅出现与扩散系数无关的一般常数，因此我们的估计对于扩散系数的跃迁是完全鲁棒的。特别地，没有施加像沿着网格顶点周围的路径单调增加那样的扩散系数的条件，因此，本结果还包括具有奇异解的情况。对于能量误差设置，关键要求变成了与原始单纯网格的顶点相关联的双单元上的扩散系数是分段恒定的，并且该方案中使用了谐波平均。这是常见的情况，例如，在我们的分析中包括以单元为中心的有限体积方法以及以顶点为中心的有限体积，有限差分和连续分段仿射有限元模型。对于双重规范设置，不需要这样的要求。我们的估计基于对H（div）的通量重构，这要归功于对双网格的所有研究方法的局部保守性，我们在本文中还提到过；还回顾了不同方法之间的相互关系。进行了数值实验，以确认导出的估计量的有保证的上限，完全鲁棒性和出色的效率。

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