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Primal-Dual Mixed Finite Element Methods for the Elliptic Cauchy Problem

Burman, E; Larson, MG; Oksanen, L; (2018) Primal-Dual Mixed Finite Element Methods for the Elliptic Cauchy Problem. SIAM Journal on Numerical Analysis , 56 (6) pp. 3480-3509. 10.1137/17M1163335. Green open access

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Abstract

We 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 Dard´e, Hannukainen, and Hyv¨onen in [SIAM J. 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.

Type: Article
Title: Primal-Dual Mixed Finite Element Methods for the Elliptic Cauchy Problem
Open access status: An open access version is available from UCL Discovery
DOI: 10.1137/17M1163335
Publisher version: https://doi.org/10.1137/17M1163335
Language: English
Additional information: This version is the version of record. For information on re-use, please refer to the publisher’s terms and conditions.
Keywords: inverse problem, elliptic Cauchy problem, mixed finite element method, primaldual method, stabilized methods
UCL classification: UCL
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics
URI: https://discovery.ucl.ac.uk/id/eprint/10060887
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