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Explicit-in-time goal-oriented adaptivity

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.

Journal Article Type Article
Publication Date Apr 15, 2019
Print ISSN 0045-7825
Electronic ISSN 1879-2138
Publisher Elsevier
Peer Reviewed Peer Reviewed
APA6 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, doi:10.1016/j.cma.2018.12.028
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
Publisher URL https://www.sciencedirect.com/science/article/pii/S0045782518306273

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