A novel penalty-based least-squares discontinuous Galerkin (LS-DG) method for time-dependent Stokes equations

In this paper, a new penalty-based least-squares discontinuous Galerkin (LS-DG) framework is proposed and analyzed for the time-dependent Stokes system. A key feature of the scheme is the use of a penalty method to address the incompressibility constraint, along with the introduction of an artificial average perturbation term in the weighted least-squares functional. This approach offers three distinct advantages: first, it facilitates the proof of coercivity of the functional, ensuring the unique solvability of the method; second, it guarantees the unconditional stability of the kinetic energy at the discrete level; and third, it integrates the strengths of both the discontinuous Galerkin and least-squares methods, naturally providing a built-in residual indicator for the adaptive procedure. We rigorously establish the well-posedness and derive optimal error estimates for the proposed scheme. Finally, the accuracy, energy stability, and high efficiency of the method are demonstrated through several numerical examples, which show that it outperforms the classical DG method in terms of computational efficiency.