Adaptive Refinement in Advection–Diffusion Problems by Anomaly Detection: A Numerical Study

Abstract

We consider advection–diffusion–reaction problems, where the advective or the reactive term is dominating with respect to the diffusive term. The solutions of these problems are characterized by the so-called layers, which represent localized regions where the gradients of the solutions are rather large or are subjected to abrupt changes. In order to improve the accuracy of the computed solution, it is fundamental to locally increase the number of degrees of freedom by limiting the computational costs. Thus, adaptive refinement, by a posteriori error estimators, is employed. The error estimators are then processed by an anomaly detection algorithm in order to identify those regions of the computational domain that should be marked and, hence, refined. The anomaly detection task is performed in an unsupervised fashion and the proposed strategy is tested on typical benchmarks. The present work shows a numerical study that highlights promising results obtained by bridging together standard techniques, i.e., the error estimators, and approaches typical of machine learning and artificial intelligence, such as the anomaly detection task.

Keywords

• advection–diffusion problems
• adaptive refinement
• a posteriori error estimates
• marking strategy
• anomaly detection