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Abstract In this paper, we propose and analyze a finite-element method of variational data assimilation for a second-order parabolic interface equation on a two-dimensional bounded domain. The Tikhonov regularization plays a key role in translating the data assimilation problem into an optimization problem. Then the existence, uniqueness and stability are analyzed for the solution of the optimization problem. We utilize the finite-element method for spatial discretization and backward Euler method for the temporal discretization. Then based on the Lagrange multiplier idea, we derive the optimality systems for both the continuous and the discrete data assimilation problems for the second-order parabolic interface equation. The convergence and the optimal error estimate are proved with the recovery of Galerkin orthogonality. Moreover, three iterative methods, which decouple the optimality system and significantly save computational cost, are developed to solve the discrete time evolution optimality system. Finally, numerical results are provided to validate the proposed method.
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An Eulerian finite element method for the linearized Navier–Stokes problem in an evolving domain
Abstract The paper addresses an error analysis of an Eulerian finite element method used for solving a linearized Navier–Stokes problem in a time-dependent domain. In this study, the domain’s evolution is assumed to be known and independent of the solution to the problem at hand. The numerical method employed in the study combines a standard backward differentiation formula-type time-stepping procedure with a geometrically unfitted finite element discretization technique. Additionally, Nitsche’s method is utilized to enforce the boundary conditions. The paper presents a convergence estimate for several velocity–pressure elements that are inf-sup stable. The estimate demonstrates optimal order convergence in the energy norm for the velocity component and a scaled $$L^{2}(H^{1})$$-type norm for the pressure component.
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- PAR ID:
- 10506590
- Publisher / Repository:
- Oxford University Press
- Date Published:
- Journal Name:
- IMA Journal of Numerical Analysis
- ISSN:
- 0272-4979
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1093/imanum/drad105
