Explicit-in-time goal-oriented adaptivity

Muñoz-Matute, Judit; Calo, Victor M.; Pardo, David; Alberdi, Elisabete; Van Der Zee, Kristoffer

Abstract

Goal-oriented adaptivity is a powerful tool to accurately approximate physically relevant solution features for Partial Differential Equations. In time dependent problems, we seek to represent the error in the quantity of interest as an integral over the whole space-time domain. A full space-time variational formulation allows such representation. Most authors employ implicit time marching schemes to perform goal-oriented adaptivity as it is known that they can be reinterpreted as Galerkin methods. In this work, we consider variational forms for explicit methods in time. We derive an appropriate error representation and propose a goal-oriented adaptive algorithm in space. For that, we derive the Forward Euler method in time employing a discontinuous-in-time Petrov-Galerkin formulation. In terms of time domain adaptivity, we impose the Courant-Friedrichs-Lewy condition to ensure the stability of the method. We provide some numerical results applied to the diffusion and the advection-diffusion equations to show the performance of the proposed explicit-in-time goal-oriented adaptive algorithm.

Citation

Muñoz-Matute, J., Calo, V. M., Pardo, D., Alberdi, E., & Van Der Zee, K. (2019). Explicit-in-time goal-oriented adaptivity. Computer Methods in Applied Mechanics and Engineering, 347, 176-200. https://doi.org/10.1016/j.cma.2018.12.028

• Journal Article Type: Article
• Acceptance Date: Dec 15, 2018
• Online Publication Date: Dec 31, 2018
• Publication Date: Apr 15, 2019
• Deposit Date: Dec 18, 2018
• Publicly Available Date: Jan 1, 2020
• Journal: Computer Methods in Applied Mechanics and Engineering
• Print ISSN: 0045-7825
• Electronic ISSN: 1879-2138
• Publisher: Elsevier
• Peer Reviewed: Peer Reviewed
• Volume: 347
• Pages: 176-200
• DOI: https://doi.org/10.1016/j.cma.2018.12.028
• Keywords: linear advection-diffusion equation; goal-oriented adaptivity; explicit methods in time; error representation; Finite Element Method
• Public URL: https://nottingham-repository.worktribe.com/output/1422422
• Publisher URL: https://www.sciencedirect.com/science/article/pii/S0045782518306273
• Additional Information: This article is maintained by: Elsevier; Article Title: Explicit-in-time goal-oriented adaptivity; Journal Title: Computer Methods in Applied Mechanics and Engineering; CrossRef DOI link to publisher maintained version: https://doi.org/10.1016/j.cma.2018.12.028; Content Type: article; Copyright: © 2018 Elsevier B.V. All rights reserved.
• Contract Date: Dec 18, 2018