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Primal-Dual Mixed Finite Element Methods for the Elliptic Cauchy Problem
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
2018 (English)In: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 56, no 6, p. 3480-3509Article in journal (Refereed) Published
Abstract [en]

consider primal-dual mixed finite element methods for the solution of the elliptic Cauchy problem, or other related data assimilation problems. The method has a local conservation property. We derive a priori error estimates using known conditional stability estimates and determine the minimal amount of weakly consistent stabilization and Tikhonov regularization that yields optimal convergence for smooth exact solutions. The effect of perturbations in data is also accounted for. A reduced version of the method, obtained by choosing a special stabilization of the dual variable, can be viewed as a variant of the least squares mixed finite element method introduced by Darde, Hannukainen, and Hyvonen in [SIAM T. Numer. Anal., 51 (2013), pp. 2123-2148]. The main difference is that our choice of regularization does not depend on auxiliary parameters, the mesh size being the only asymptotic parameter. Finally, we show that the reduced method can be used for defect correction iteration to determine the solution of the full method. The theory is illustrated by some numerical examples.

Place, publisher, year, edition, pages
SIAM Publications , 2018. Vol. 56, no 6, p. 3480-3509
Keywords [en]
inverse problem, elliptic Cauchy problem, mixed finite element method, primal-dual method, stabilized methods
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:umu:diva-155126DOI: 10.1137/17M1163335ISI: 000453787700013OAI: oai:DiVA.org:umu-155126DiVA, id: diva2:1276503
Funder
Swedish Foundation for Strategic Research , AM13-0029Swedish Research Council, 2013-04708Swedish Research Council, 2017-03911eSSENCE - An eScience CollaborationAvailable from: 2019-01-08 Created: 2019-01-08 Last updated: 2019-01-08Bibliographically approved

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Larson, Mats G.

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