# On three steps two-grid finite element methods for the 2D-transient Navier-Stokes equations

Saumya Bajpai and Amiya K. Pani

# Abstract

In this paper, an error analysis of a three steps two level Galerkin finite element method for the two dimensional transient Navier–Stokes equations is discussed. First of all, the problem is discretized in spatial direction by employing finite element method on a coarse mesh 𝒯H with mesh size H. Then, in step two, the nonlinear system is linearized around the coarse grid solution, say, uH, which is similar to Newton’s type iteration and the resulting linear system is solved on a finer mesh 𝒯h with mesh size h. In step three, a correction is obtained through solving a linear problem on the finer mesh and an updated final solution is derived. Optimal error estimates in L(L2)-norm, when h = 𝒪 (H2−δ) and in L(H1)-norm, when h = 𝒪(H4−δ) for the velocity and in L(L2)-norm, when h = 𝒪(H4−δ) for the pressure are established for arbitrarily small δ > 0. Further, under uniqueness assumption, these estimates are proved to be valid uniformly in time. Then, based on backward Euler method, a completely discrete scheme is analyzed and a priori error estimates are derived. Results obtained in this paper are sharper than those derived earlier by two-grid methods. Finally, the paper is concluded with some numerical experiments.

MSC 2010: 65M60; 65M12; 65M15; 35D05; 35D10

# Acknowledgment

The authors would like to thank the referees for their valuable suggestions. The second author acknowledges the support provided by the National Programme on Differential Equations: Theory, Computation and Applications (NPDE-TCA) under the DST project No.SR/S4/MS:639/90.

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