Maximum norm a posteriori error estimates for convection-diffusion problems

被引:1
作者
Demlow, Alan [1 ]
Franz, Sebastian [2 ]
Kopteva, Natalia [3 ]
机构
[1] Texas A&M Univ, Dept Math, College Stn, TX 77843 USA
[2] Tech Univ Dresden, Inst Sci Comp, D-01217 Dresden, Germany
[3] Univ Limerick, Dept Math & Stat, Limerick V94 T9PX, Ireland
基金
爱尔兰科学基金会; 美国国家科学基金会;
关键词
a posteriori error estimate; maximum norm; singular perturbation; convection-diffusion; GALERKIN APPROXIMATIONS; EQUATIONS; STABILIZATION;
D O I
10.1093/imanum/drad001
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
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 Verfurth 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.
引用
收藏
页码:2562 / 2584
页数:23
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