Abstract
We consider data assimilation for the heat equation using a finite element space semi-discretization. The approach is optimization based, but the design of regularization operators and parameters rely on techniques from the theory of stabilized finite elements. The space semi-discretized system is shown to admit a unique solution. Combining sharp estimates of the numerical stability of the discrete scheme and conditional stability estimates of the ill-posed continuous pde-model we then derive error estimates that reflect the approximation order of the finite element space and the stability of the continuous model. Two different data assimilation situations with different stability properties are considered to illustrate the framework. Full detail on how to adapt known stability estimates for the continuous model to work with the numerical analysis framework is given in “Appendix”.
Highlights
IntroductionAn important feature of the present work is that the choice of the regularizing terms is driven by the analysis and designed to give error estimates that reflect the approximation properties of the finite element space and the stability of the continuous model
We consider two data assimilation problems for the heat equation∂t u − u = f, in (0, T ) ×, (1)E
Let ω, B ⊂ be open and let 0 < T1 < T2 ≤ T. Both the data assimilation problems are of the following general form: determine the restriction u|(T1,T2)×B of a solution to the heat equation (1) given f and the restriction u|(0,T )×ω
Summary
An important feature of the present work is that the choice of the regularizing terms is driven by the analysis and designed to give error estimates that reflect the approximation properties of the finite element space and the stability of the continuous model. Under suitable choices of the semi-discrete spaces and the regularization, we show that (6) has a unique minimizer (uh, zh), h > 0, satisfying the following convergence rate uh − u L2(T1,T2;H1(B)) ≤ C hκ ( u ∗ + f ),. Where u is the unique solution of the continuum data assimilation problem, κ ∈ (0, 1) is the constant in Theorem 1, and the norms on the right-hand side capture the assumed apriori smoothness, see Theorem 3 below for the precise formulation. We provide a computational study of such rescaling in [10]
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