Goal‐oriented adaptivity using unconventional error representations for the multidimensional Helmholtz equation
Year: 2017
Type: article
Abstract: Summary In goal‐oriented adaptivity, the error in the quantity of interest is represented using the error functions of the direct and adjoint problems. This error representation is subsequently bounded above by element‐wise error indicators that are used to drive optimal refinements. In this work, we propose to replace, in the error representation, the adjoint problem by an alternative operator. The main advantage of the proposed approach is that, when judiciously selecting such alternative operator, the corresponding upper bound of the error representation becomes sharper, leading to a more efficient goal‐oriented adaptivity. While the method can be applied to a variety of problems, we focus here on two‐ and three‐dimensional (2‐D and 3‐D) Helmholtz problems. We show via extensive numerical experimentation that the upper bounds provided by the alternative error representations are sharper than the classical ones and lead to a more robust p‐adaptive process. We also provide guidelines for finding operators delivering sharp error representation upper bounds. We further extend the results to a convection‐dominated diffusion problem as well as to problems with discontinuous material coefficients. Finally, we consider a sonic logging‐while‐drilling problem to illustrate the applicability of the proposed method.
DOI: 10.1002/nme.5601
Institutions: Institut national de recherche en sciences et technologies du numérique, University of the Basque Country, Centre Inria de l'université de Bordeaux, Basque Center for Applied Mathematics, Pontificia Universidad Católica de Valparaíso, Ikerbasque, South Dakota School of Mines and Technology, Polytechnique Montréal
Language: English
Subfield: Computational Mechanics
Field: Engineering
Domain: Physical Sciences
Open Access status: green
Funders: European Commission, Eusko Jaurlaritza, Ministerio de Economía y Competitividad, Ministerio de Ciencia y Tecnología, Basque Center for Applied Mathematics, Natural Sciences and Engineering Research Council of Canada, Agencia Estatal de Investigación, Fondo Nacional de Desarrollo Científico y Tecnológico
Awards: G1029768446, G1111619560, G1708820433, G3013138087, G3244436707, G3490360365, G3743541819, G3779338858, G4252991836, Geophysical Exploration using Advanced GAlerkin Methods, G434337789, G4373130462, G5149605805, G5281057304, G5291586380, G6649204693, G7092330731, G711474080, G8296132195, G862438431, G8921874968
OpenAlex ID: W2733489173