Analysis of a class of globally divergence-free linearized fully discrete WG methods for unsteady Brinkman-Forchheimer equations

In this paper, we develop and analyze a class of linearized fully discrete weak Galerkin (WG) finite element methods for the unsteady Brinkman-Forchheimer model. For spatial discretization, the WG methods utilize the piecewise polynomials of degrees m ( m ≥ 1) and m − 1 to approximate the velocity and the pressure within the interior of elements, respectively. Additionally, piecewise polynomials of degree m are employed for their corresponding numerical traces on the element interfaces. For temporal discretization, a linearized backward Euler difference scheme is formulated, where semi-implicit techniques are adopted to deal with the nonlinear terms. The proposed methods ensure that the velocity approximation is globally divergence - free. The well-posedness and optimal error estimates, both in the energy norm and L 2 norm, are rigorously derived. Numerical tests are conducted to validate the established theoretical results.