Analysis of continuous data assimilation with large (or even infinite) nudging parameters

TL;DR

This work addresses the challenge of achieving long-time, optimally accurate solutions in continuous data assimilation (CDA) when the nudging parameter $\\mu$ is allowed to be arbitrarily large. By introducing CDA Poisson and CDA Stokes projections, the authors establish projection-based error estimates that are independent of $\\mu$ and apply them to a BDF2–FEM discretization of the heat equation and the Navier–Stokes equations, proving mu-free long-time $L^2$ accuracy under suitable conditions. The approach is validated with extensive numerical experiments across heat, transport, NSE, and Cahn–Hilliard problems, demonstrating exponential convergence in time up to discretization error and robust mu-insensitive long-time behavior. The results justify using large or even direct Dirichlet enforcement nudging in practice, offering a solid theoretical foundation for CDA methods in complex dissipative PDEs and guiding future work on noisy data scenarios.

Abstract

This paper considers continuous data assimilation (CDA) in partial differential equation (PDE) discretizations where nudging parameters can be taken arbitrarily large. We prove that long-time optimally accurate solutions are obtained for such parameters for the heat and Navier-Stokes equations (using implicit time stepping methods), with error bounds that do not grow as the nudging parameter gets large. Existing theoretical results either prove optimal accuracy but with the error scaled by the nudging parameter, or suboptimal accuracy that is independent of it. The key idea to the improved analysis is to decompose the error based on a weighted inner product that incorporates the (symmetric by construction) nudging term, and prove that the projection error from this weighted inner product is optimal and independent of the nudging parameter. We apply the idea to BDF2 - finite element discretizations of the heat equation and Navier-Stokes equations to show that with CDA, they will admit optimal long-time accurate solutions independent of the nudging parameter, for nudging parameters large enough. Several numerical tests are given for the heat equation, fluid transport equation, Navier-Stokes, and Cahn-Hilliard that illustrate the theory.

Figures (8)

• Figure 1: Shown above are $L^2$ error versus time for the tests using the analytical solution for the heat equation and BDF2-CDA algorithm, for varying $h$ (left) and varying $\Delta t$ (right).
• Figure 2: Shown above are the fine mesh used for the finite element computations (left) and the coarse mesh nodes used for CDA (right).
• Figure 3: Shown above are plots of convergence of the BDF2-CDA solutions to the true solution, for varying $\mu$.
• Figure 4: Contour plots of DA and DNS velocity magnitudes at times 0, 1, 2 and 5.
• Figure 5: Shown above are convergence plots of the CDA solution to the true solution for the Kelvin-Helmholtz test problem.

Theorems & Definitions (12)

• Lemma 2.1
• Lemma 3.1
• proof
• Lemma 3.2
• Lemma 3.3
• proof
• Theorem 4.1
• Remark 4.2
• proof
• Theorem 5.1