A New Variational Multiscale FEM for the Steady-State Natural Convection Problem with Bubble Stabilization

Abstract

In this paper, a new variational multiscale finite element method (FEM) with bubble stabilization for the steady-state natural convection problem is presented. The new variational multiscale FEM is derived by using a decoupling and linear technique. Compared with the common projection-based stabilized FEM, the new method does not need to introduce any extra degree of freedom. Moreover, the new method has the same convergence precision as the standard Galerkin FEM. Finally, some numerical examples are given to show that the new method is efficient, reliable, and can save a lot of CPU-time for this problem.

The characteristic variational multiscale method for time dependent conduction–convection problems

“…[3]). A new VMS method is presented for steady-state natural convection problem with bubble stabilized FEM in [18]. In this paper, we shall use the C-VMS method to solve the nonstationary conduction-convection problems.…”

An energetically consistent concurrent multiscale method for heterogeneous heat transfer and phase transition applications

“…The coarse scale is then projected into an appropriate subspace while the fine scale is directly stabilized against highly localized affects such as those seen in turbulence modeling. This method has been further developed in later work and applied to problems involving incompressible flows [21,22,23] and convection-diffusion [24,25,26].…”

Implicit–explicit schemes of finite element method for the non-stationary thermal convection problems with temperature-dependent coefficients

“…The scheme of (8) consists of two subproblems: one is a linearized convection-diffusion equation and the other is a linearized Navier-Stokes equations. First we can solve the second equation of (8) and obtain θ n h , and then solve the first equation of (8) to obtain (u n h , p n h ).…”

“…Next we state the main results of this paper for above two schemes. (2) and (8), respectively. Then there hold that…”

“…Thus, development of an efficient computational method for investigating this model has drawn the attentions of many researchers because of many real applications. There are so far numerous works devoted to the development of efficient numerical schemes for the conduction-convection equations with constant coefficients [1][2][3][4][5][6][7][8]. For example, the finite element method, the penalty method, the variational multiscale method, etc.…”