Finite element method in the solution of the Euler and Navier‐Stokes equations for internal flow

Abstract

Finite element solutions of the Euler and Navier-Stokes equations are presented, using a simple dissipation model. The discretization is based on the weak-Galerkin weighted residual method and equal interpolation functions for all the unknowns are permitted. The nonlinearity is iterated upon using a Newton method and at each iteration the linear algebraic system is solved by a direct solver with all unknowns fully coupled. Results are presented for two-dimensional transonic inviscid flows and two-and three-dimensional incompressible viscous flows. Convergence of the algorithm is shown to be quadratic, reaching machine accuracy in very few iterations. The inviscid results demonstrate the existence of nonunique numerical solutions to the steady Euler equations.

A stabilised nonconforming finite element method for steady incompressible flows

“…Thus, development of an efficient and effective computational method for investigating the incompressible flow problem has practical significance, and has drawn the attention of many researchers (see Habashi et al 1991, Makihara et al 1999, Scott and Chang 1995, Picasso and Rappaz 2001, Xu 2001, Ye 2006, Liu and Hou 2009, Shi and Ren 2009, Aliabadi et al 2011, Huang et al 2011, and the references cited therein).…”