A three-step, two-grid method, in combination with the discontinuous Galerkin (DG) method, is applied and analyzed for the Kelvin-Voigt viscoelastic fluid flow model. We employ the DG method to solve the nonlinear system on a coarse grid \(\mathcal{E}_H\) with grid size \(H\) , in the first step, whereas in the second step, a linearized DG scheme, using Newton’s one step, is employed on a fine grid \(\mathcal{E}_h\) with size \(h\ll H\) . The third and final step, which is a correction step, delivers a modified final solution through an improvised linear DG scheme on the fine grid. A priori error estimates of an optimal nature are presented for the appropriate scaling of the grid parameters, which also demonstrates the efficiency of the scheme. For example, for optimal velocity error in energy norm and pressure error in \(L^2\) -norm, we have \(h\approx \mathcal{O}(H^5)\) . Subsequently, a fully discrete two-grid DG scheme is considered, employing the first-order backward Euler method for time discretization, and optimal fully discrete error estimates are derived. Our calculations rely on newly derived interpolated Sobolev and trace inequalities, which enable optimal estimation of the convection-related terms. The article concludes by discussing a few numerical results that demonstrate our theoretical findings as well as highlight the computational efficiency, achieving nearly half the computational time of the standard DG method.
