A nonconforming finite element method for data assimilation subject to the transient Stokes problem

In this study, we will consider the unique continuation problem for reconstructing the final state of the transient Stokes problem when the initial data is unknown, but additional data is given in a subdomain in space-time. The backward differentiation method is used to discretise the time derivativ...

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Bibliographic Details
Published in:Numerische Mathematik Vol. 157; no. 6; pp. 2017 - 2054
Main Authors: Burman, Erik, Garg, Deepika, Preuss, Janosch
Format: Journal Article
Language:English
Published: Heidelberg Springer Nature B.V 01.12.2025
Subjects:
Discrete systems
Finite element method
ISSN:0029-599X, 0945-3245
Online Access:Get full text
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Summary:In this study, we will consider the unique continuation problem for reconstructing the final state of the transient Stokes problem when the initial data is unknown, but additional data is given in a subdomain in space-time. The backward differentiation method is used to discretise the time derivative and standard nonconforming affine finite element approximation is applied for the discretisation in space. The discrete system is regularized by adding a penalty of the $$H^1$$ H 1 -semi-norm of the initial data, scaled with the mesh parameter. The scaling is chosen so that an optimal error estimate holds in $$L^2(T_1,T;H^1(\Omega ))$$ L 2 ( T 1 , T ; H 1 ( Ω ) ) , $$T_1>0$$ T 1 > 0 . The estimate is derived using the Lipschitz stability of the reconstruction problem and interpolation between discrete spaces. The theory is validated on some numerical examples.
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ISSN:0029-599X
0945-3245
DOI:10.1007/s00211-025-01500-8