In this work we propose and investigate the performance of a metric-based refinement criteria for adaptive meshing used for improving the numerical solution of an elliptic problem. We show that in general, when solving elliptic equations such as the Poisson-Helmholtz equation, the minimization of the interpolation error often used as local refinement criteria does not always guarantee the minimization of the total numerical error. Numerical and theoretical arguments are given to unveil the critical role of the mesh compression – the size aspect ratio between the finest cell size and the mean cell size of an adapted mesh – to determine whether the estimated error is purely local meaning that the interpolation error is a good enough error model for the total error or if other, non-local, sources of error need to be accounted for. We show through particular examples that a slightly sub-optimal mesh in terms of interpolation error may significantly reduce the total error of a numerical solution, depending on the value of the compression ratio and not on the number of grid points. Based on this observation, we propose a new method to exclude the grids where non-local errors can control the accuracy of the solution. This is achieved by an automatic estimation of the optimal compression ratio, which is imposed as an additional constraint in the minimal element size during the mesh adaptation process. The method is tested on quadtree and octree grids, showing very satisfactory performances in reducing the total numerical error despite the additional constrain imposed.

中文翻译：

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一种基于度量的约束下自适应网格细化准则，用于解决四叉树/八叉树网格上的椭圆问题

在这项工作中，我们提出并研究了基于度量的自适应网格细化标准的性能，该标准用于改进椭圆问题的数值解。我们表明，一般来说，在求解泊松-亥姆霍兹方程等椭圆方程时，通常用作局部细化标准的插值误差的最小化并不总是保证总数值误差的最小化。给出了数值和理论论证来揭示网格压缩的关键作用（自适应网格的最细单元尺寸与平均单元尺寸之间的尺寸纵横比），以确定估计误差是否纯粹是局部的，这意味着插值误差是对于总误差或是否需要考虑其他非局部误差源，需要有一个足够好的误差模型。我们通过特定的例子表明，插值误差稍微次优的网格可能会显着降低数值解的总误差，具体取决于压缩比的值而不是网格点的数量。基于这一观察，我们提出了一种新方法来排除非局部误差可以控制解决方案准确性的网格。这是通过自动估计最佳压缩比来实现的，该压缩比是在网格自适应过程中作为最小单元尺寸的附加约束而施加的。该方法在四叉树和八叉树网格上进行了测试，尽管施加了额外的约束，但在减少总数值误差方面表现出了非常令人满意的性能。