# Stabilized Lagrange–Galerkin Schemes of First- and Second-Order in Time for the Navier–Stokes Equations

## Abstract

Two stabilized Lagrange–Galerkin schemes for the Navier–Stokes equations are reviewed. The schemes are based on a combination of the Lagrange–Galerkin method and Brezzi–Pitkäranta’s stabilization method. They maintain the advantages of both methods: (i) They are robust for convection-dominated problems and the systems of linear equations to be solved are symmetric; and (ii) Since the P1 finite element is employed for both velocity and pressure, the numbers of degrees of freedom are much smaller than that of other typical elements for the equations, e.g., P2/P1. Therefore, the schemes are efficient especially for three-dimensional problems. The one of the schemes is of first-order in time by Euler’s method and the other is of second-order by Adams–Bashforth’s method. In the second-order scheme an additional initial velocity is required. A convergence analysis is done for the choice of the velocity obtained by the first-order scheme, whose theoretical result is also recognized numerically.

## Notes

### Acknowledgements

This work was supported by JSPS (the Japan Society for the Promotion of Science) under the Japanese–German Graduate Externship (Mathematical Fluid Dynamics) and Grant-in-Aid for Scientific Research (S), No. 24224004. The authors are indebted to JSPS also for Grant-in-Aid for Young Scientists (B), No. 26800091 to the first author and for Grant-in-Aid for Scientific Research (C), No. 25400212 to the second author.

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