Error Estimates of a Stabilized Lagrange–Galerkin Scheme of Second-Order in Time for the Navier–Stokes Equations
Error estimates with optimal convergence orders are proved for a stabilized Lagrange–Galerkin scheme of second-order in time for the Navier–Stokes equations. The scheme is a combination of Lagrange–Galerkin method and Brezzi–Pitkäranta’s stabilization method. It maintains the advantages of both methods; (i) It is robust for convection-dominated problems and the system of linear equations to be solved is symmetric. (ii) Since the P1 finite element is employed for both velocity and pressure, the number of degrees of freedom is much smaller than that of other typical elements for the equations, e.g., P2/P1. Therefore, the scheme is efficient especially for three-dimensional problems. The second-order accuracy in time is realized by Adams-Bashforth’s (two-step) method for the discretization of the material derivative along the trajectory of fluid particles. The theoretical convergence orders are recognized by two- and three-dimensional numerical results.
KeywordsError estimates Stabilized Lagrange-Galerkin scheme Second-order scheme Navier-Strokes equations
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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