On a two-grid finite element scheme for the equations of motion arising in Kelvin-Voigt model

Abstract

In this paper, we study a two level method based on Newton’s iteration for the nonlinear system arising from the Galerkin finite element approximation to the equations of motion described by the Kelvin-Voigt viscoelastic fluid flow model. The two-grid algorithm is based on three steps. In the first step, the nonlinear system is solved on a coarse mesh $\mathcal {T}_{H}$ to obtain an approximate solution u _H . In the second step, the nonlinear system is linearized around u _H based on Newton’s iteration and the linear system is solved on a finer mesh $\mathcal {T}_{h}$. Finally, in the third step, a correction to the results obtained in the second step is achieved by solving a linear problem with a different right hand side on $\mathcal {T}_{h}$. Optimal error estimates in L ^∞(L ²)-norm, when $h=\mathcal {O} (H^{2-\delta })$ and in L ^∞(¹)-norm, when $h=\mathcal {O}(H^{5-2\delta })$ for velocity and in L ^∞(L ²)-norm, when $h=\mathcal {O}(H^{5-2\delta })$ for pressure are established, where δ > 0 is arbitrarily small for two dimensions and $\delta =\frac {1}{2}$ for three dimensions.