Robust globally divergence-free weak Galerkin variational data assimilation method for convection-dominated Oseen equations

Abstract

This paper presents a weak Galerkin (WG) finite element method based on the variational approach for data assimilation of the unsteady convection-dominated Oseen equation. The WG scheme uses piecewise polynomials of degrees k($k\ge 1$) and $k-1$ respectively for the approximations of the velocity and the pressure in the interior of elements, and uses piecewise polynomials of degree k for their numerical traces on the interfaces of elements. The method is shown to yield globally divergence-free approximations of the velocity and initial value. It is proved that the velocity error in the $L^2$-norm has a Reynolds-robust error bound with quasi-optimal convergence order $k+1/2$ in the convection-dominated region. To solve the discrete optimality system efficiently, the conjugate gradient iterative algorithm is developed, which also preserves the globally divergence-free property of WG scheme. Numerical experiments are provided to verify the obtained theoretical results.