An optimal nonlinear Galerkin method with mixed finite elements for the steady Navier-Stokes equations

✍ Scribed by Yinnian He; Aiwen Wang; Zhangxin Chen; Kaitai Li

• Publisher: John Wiley and Sons
• Year: 2003
• Tongue: English
• Weight: 113 KB
• Volume: 19
• Category: Article
• ISSN: 0749-159X
• DOI: 10.1002/num.10074

✦ Synopsis

Abstract

An optimal nonlinear Galerkin method with mixed finite elements is developed for solving the two‐dimensional steady incompressible Navier‐Stokes equations. This method is based on two finite element spaces X~H~ and X~h~ for the approximation of velocity, defined on a coarse grid with grid size H and a fine grid with grid size h ≪ H, respectively, and a finite element space M~h~ for the approximation of pressure. We prove that the difference in appropriate norms between the solutions of the nonlinear Galerkin method and a classical Galerkin method is of the order of H^5^. If we choose H = O(h^2/5^), these two methods have a convergence rate of the same order. We numerically demonstrate that the optimal nonlinear Galerkin method is efficient and can save a large amount of computational time. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 762–775, 2003.