A Stabilized Galerkin Scheme for the Convection-Diffusion-Reaction Equations
- PDF / 1,199,740 Bytes
- 20 Pages / 439.37 x 666.142 pts Page_size
- 15 Downloads / 201 Views
A Stabilized Galerkin Scheme for the Convection-Diffusion-Reaction Equations Qingfang Liu · Yanren Hou · Lei Ding · Qingchang Liu
Received: 25 October 2012 / Accepted: 25 July 2013 / Published online: 23 August 2013 © Springer Science+Business Media Dordrecht 2013
Abstract A fully discrete stabilized scheme is proposed for solving the time-dependent convection-diffusion-reaction equations. A time derivative term results in our stabilized algorithm. The finite element method for spatial discretization and the backward Euler or Crank-Nicolson scheme for time discretization are employed. The long-time stability and convergence are established in this article. Finally, some numerical experiments are provided to confirm the theoretical analysis. Keywords Stabilized method · Finite element method · Convection-diffusion-reaction equations · Stability · Error estimates
Subsidized by the Fundamental Research Funds for the Central Universities (Grant Nos. 08142013 and 08143045), NSF of China (Grant Nos. 11171269 and 11201254) and the Ph.D. Programs Foundation of Ministry of Education of China (Grant No. 20110201110027).
B
Q. Liu · Y. Hou ( ) School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, P.R. China e-mail: [email protected] Q. Liu e-mail: [email protected] Q. Liu · Y. Hou Center for Computational Geosciences, Xi’an Jiaotong University, Xi’an 710049, P.R. China L. Ding School of Electronic and Information Engineering, Xi’an Jiaotong University, Xi’an 710049, P.R. China e-mail: [email protected] Q. Liu School of Mechanics and Civil & Architecture, Northwestern Polytechnical University, Xi’an 710129, P.R. China e-mail: [email protected]
116
Q. Liu et al.
1 Introduction We consider the time-dependent convection-diffusion-reaction equation ⎧ ⎨ ut − νu + b · ∇u + cu = f, ∀(x, t) ∈ Ω × [0, T ], ∀x ∈ Ω, u(x, 0) = u0 , ⎩ u(x, t) = 0, ∀(x, t) ∈ ∂Ω × [0, T ].
(1.1)
Here Ω is a bounded domain in R m (m = 1, 2, 3) with a Lipschitz continuous boundary ∂Ω, u(x, t) is a scalar function representing some quality of a fluid flow such as temperature or contaminant level, ν is a positive constant which is called the diffusion coefficient, b(x, t) ∈ L∞ (0, T ; L∞ (Ω))m and c(x, t) ∈ L∞ (0, T ; L∞ (Ω)) are given functions, f ∈ L2 (0, T ; L2 (Ω)) is a forcing function and T represents a finite time. During the last two decades, various numerical methods for solving the convectiondiffusion-reaction type equations have been studied (e.g., see the work of Codina [1], Hughes et al. [2–4], John and Kaya [5, 6], Layton et al. [7–14], Guermond [15, 16], Heitmann [17], Santos and Almeida [18], Davis and Pahlevani [19]). In particular, a large number of works have been devoted to the research of stabilized methods (see Codina [1] for a survey of some most popular methods). An alternative stabilized technique is the so-called artificial viscosity methods (AV) which add a suitable artificial viscosity term as a stabilized factor. For a given mesh scale h > 0, we define a finite element subspace Vh
Data Loading...
