The weak Galerkin finite element method for incompressible flow

We study the weak Galerkin finite element method for stationary Navier-Stokes problem. We propose a weak finite element velocity-pressure space pair that satisfies the discrete inf-sup condition. This space pair is then employed to construct a stable weak Galerkin finite element scheme without adding any stabilizing term or penalty term. We prove a discrete embedding inequality on the weak finite element space which, together with the discrete inf-sup condition, enables us to establish the unique existence and stability estimates of the discrete velocity and pressure. Then, we derive the optimal error estimates for velocity and pressure approximations in the H1-norm and L2-norm, respectively. Numerical experiments are provided to illustrate the theoretical analysis.