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 Verf\"{u}rth 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.
翻译:我们证明,在线性天平椭圆对流问题的最大规范中,我们是一种事后误差估计,这种误差可能是极不稳定的。Verf\"{u}rth在能源规范中的类似误差分析表明,自然能源规范中必须加上{相偏差衍生物的双重规范,以使自然残留估计器可靠和高效。我们显示,这种情况与最高规范相似。特别是,我们定义了对流错误的网状依赖加权半调值,该误差作为能源规范设置中所使用的双规范的最大对等值。总误差随后被定义为这个半调值的总和、误差的最大规范以及数据振荡。自然最高常态残留误差估计值显示与这个总误差概念相当,且始终独立于单度扰动参数。这些估计是在以下假设下得到证明的:在手头上的绿色函数中,某些自然估计具有对问题的最大对等值。数字实验后,总误判实验将有效测量到迷性摄测层中,从而有效测量了我们最强的定位。