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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| Vydáno v: | Numerische Mathematik Ročník 157; číslo 6; s. 2017 - 2054 |
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01.12.2025
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| Abstract | 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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| AbstractList | 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. 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 H1-semi-norm of the initial data, scaled with the mesh parameter. The scaling is chosen so that an optimal error estimate holds in L2(T1,T;H1(Ω)), T1>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. |
| Author | Garg, Deepika Burman, Erik Preuss, Janosch |
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| Cites_doi | 10.1007/978-3-642-22980-0 10.1090/mcom/3092 10.1137/060670961 10.1007/s007910050004 10.1137/22M1542933 10.5802/smai-jcm.122 10.1137/16M110962X 10.1051/m2an/2023106 10.1007/s10444-020-09806-x 10.1051/m2an/2020062 10.1007/978-1-4757-4355-5 10.1137/22M1508637 10.1007/s00211-018-0970-6 10.1051/m2an/2010058 10.1137/130916862 10.1090/mcom/3255 10.1007/978-0-387-75934-0 10.1016/j.cma.2020.113224 10.1051/m2an/2018030 10.1088/1361-6420/ab9161 10.1007/s00211-018-0949-3 10.1090/S0025-5718-07-01951-5 10.1090/mcom/3240 10.1007/s00332-013-9189-y |
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| Title | A nonconforming finite element method for data assimilation subject to the transient Stokes problem |
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