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 TH 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 Th. 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 Th. Optimal error estimates in L ∞(L 2)-norm, when h=O(H2−δ) and in L ∞(1)-norm, when h=O(H5−2δ) for velocity and in L ∞(L 2)-norm, when h=O(H5−2δ) for pressure are established, where δ > 0 is arbitrarily small for two dimensions and δ=12 for three dimensions.