论文标题
最大规范A后验错误估计对流扩散问题
Maximum norm a posteriori error estimates for convection-diffusion problems
论文作者
论文摘要
我们证明，对于可能会奇异的扰动的线性标量椭圆对流 - 扩散问题，在最大规范的后验误差估计中是残留类型的。 Verfürth对能量规范的类似误差分析表明，必须将{对流衍生物的双重标准添加到自然能量规范中，以使自然残留估计器可靠和有效。我们表明，最大规范的情况相似。特别是，我们定义了对流误差的网状加权符号，该误差是与能量规范设置中使用的双重标准的最大值对应物。然后将总误差定义为此eminorm的总和，误差的最大规范和数据振荡。自然最大规范残留误差估计器被证明等同于此总误差概念，恒定与单数扰动参数无关。这些估计是在假设的假设中证明了某些自然估计值对绿色的功能而言是手头问题的功能。数值实验证实，我们的估计器有效地捕获了奇异扰动问题的最大声称误差行为，并且可以有效地驱动自适应改进以捕获层现象。
We prove residual-type a posteriori error estimates in the maximum norm for a linear scalar elliptic convection-diffusion problem that may be singularly perturbed. Similar error analysis in the energy norm by Verfürth indicates that a dual norm of the {convective derivative of the} error must be added to the natural energy norm in order for the natural residual estimator to be reliable and efficient. We show that the situation is similar for the maximum norm. In particular, we define a mesh-dependent weighted seminorm of the convective error which functions as a maximum-norm counterpart to the dual norm used in the energy norm setting. The total error is then defined as the sum of this seminorm, the maximum norm of the error, and data oscillation. The natural maximum norm residual error estimator is shown to be equivalent to this total error notion, with constant independent of singular perturbation parameters. These estimates are proved under the assumption that certain natural estimates hold for the Green's function for the problem at hand. Numerical experiments confirm that our estimators effectively capture the maximum-norm error behavior for singularly perturbed problems, and can effectively drive adaptive refinement in order to capture layer phenomena.
