We consider second-order PDE problems set in unbounded domains and discretized by Lagrange finite elements on a finite mesh, thus introducing an artificial boundary in the discretization. Specifically, we consider the reaction diffusion equation as well as Helmholtz problems in waveguides with perfectly matched layers. The usual procedure to deal with such problems is to first consider a modeling error due to the introduction of the artificial boundary, and estimate the remaining discretization error with a standard a posteriori technique. A shortcoming of this method, however, is that it is typically hard to obtain sharp bounds on the modeling error. In this work, we propose a new technique that allows to control the whole error by an a posteriori error estimator. Specifically, we propose a flux-equilibrated estimator that is slightly modified to handle the truncation boundary. For the reaction diffusion equation, we obtain fully-computable guaranteed error bounds, and the estimator is locally efficient and polynomial-degree-robust provided that the elements touching the truncation boundary are not too refined. This last condition may be seen as an extension of the notion of shape-regularity of the mesh, and does not prevent the design of efficient adaptive algorithms. For the Helmholtz problem, as usual, these statements remain valid if the mesh is sufficiently refined. Our theoretical findings are completed with numerical examples which indicate that the estimator is suited to drive optimal adaptive mesh refinements.

翻译：本文研究在无界域中设置、并通过有限网格上的拉格朗日有限元进行离散化的二阶偏微分方程问题，由此在离散化过程中引入了人工边界。具体而言，我们考虑反应扩散方程以及带有完美匹配层的波导中的亥姆霍兹问题。处理此类问题的常规方法是：首先考虑由人工边界引入的建模误差，然后采用标准后验技术估计剩余的离散化误差。然而，该方法的缺陷在于通常难以获得建模误差的严格界。本工作中，我们提出一种新技术，能够通过后验误差估计量控制整体误差。具体来说，我们提出一种经过微调以处理截断边界的通量平衡估计量。对于反应扩散方程，我们获得了完全可计算的保证误差界，且该估计量具有局部高效性和多项式次数鲁棒性——前提是接触截断边界的单元未过度细化。最后这一条件可视为网格形状正则性概念的扩展，且不妨碍高效自适应算法的设计。对于亥姆霍兹问题，与常规情况相同，若网格充分细化，这些结论依然成立。我们通过数值算例完善了理论发现，结果表明该估计量适用于驱动最优的自适应网格细化。