Monotonicity recovering and accuracy preserving optimization methods for postprocessing finite element solutions

JOURNAL OF COMPUTATIONAL PHYSICS

卷:231

Article

Burdakov, Oleg¹  Kapyrin, Ivan²  Vassilevski, Yuri² 

[1] Linkoping Univ, Dept Math, S-58183 Linkoping, Sweden

[2] Russian Acad Sci, Inst Numer Math, Moscow, Russia

关键词: Finite element solution;    Accuracy analysis;    Constrained monotonic regression;    Large scale quadratic optimization;    Lagrangian relaxation;    Dual ascent method;   

DOI  :  10.1016/j.jcp.2011.12.041

来源: Elsevier

【 摘 要 】

We suggest here a least-change correction to available finite element (FE) solution. This postprocessing procedure is aimed at recovering the monotonicity and some other important properties that may not be exhibited by the FE solution. Although our approach is presented for FEs, it admits natural extension to other numerical schemes, such as finite differences and finite volumes. For the postprocessing, a priori information about the monotonicity is assumed to be available, either for the whole domain or for a subdomain where the lost monotonicity is to be recovered. The obvious requirement is that such information is to be obtained without involving the exact solution, e.g. from expected symmetries of this solution. The postprocessing is based on solving a monotonic regression problem with some extra constraints. One of them is a linear equality-type constraint that models the conservativity requirement. The other ones are box-type constraints, and they originate from the discrete maximum principle. The resulting postprocessing problem is a large scale quadratic optimization problem. It is proved that the postprocessed FE solution preserves the accuracy of the discrete FE approximation. We introduce an algorithm for solving the postprocessing problem. It can be viewed as a dual ascent method based on the Lagrangian relaxation of the equality constraint. We justify theoretically its correctness. Its efficiency is demonstrated by the presented results of numerical experiments. (C) 2012 Elsevier Inc. All rights reserved.

【 授权许可】

Free