scholarly journals An embedded discontinuous Galerkin method for the Oseen equations

2021 ◽  
Author(s):  
Yongbin Han ◽  
Yanren Hou
Keyword(s):  
Galerkin Method ◽  
Discontinuous Galerkin ◽  
Error Bound ◽  
Discontinuous Galerkin Method ◽  
Convergence Order ◽  
Optimal Error Estimate ◽  
Optimal Error ◽  
Oseen Equations ◽  
Optimal Error Bound ◽  
Convection Dominated

In this paper, the a prior error estimates of an embedded discontinuous Galerkin method for the Oseen equations are presented. It is proved that the velocity error in the L 2 (Ω) norm, has an optimal error bound with convergence order k + 1, where the constants are dependent on the Reynolds number (or ν − 1 ), in the diffusion-dominated regime, and in the convection-dominated regime, it has a Reynolds-robust error bound with quasi-optimal convergence order k +1 / 2. Here, k is the polynomial order of the velocity space. In addition, we also prove an optimal error estimate for the pressure. Finally, we carry out some numerical experiments to corroborate our analytical results.

A Sub-Grid Structure Enhanced Discontinuous Galerkin Method for Multiscale Diffusion and Convection-Diffusion Problems

2013 ◽  
Vol 14 (2) ◽  
pp. 370-392 ◽  
Author(s):  
Eric T. Chung ◽  
Wing Tat Leung
Keyword(s):  
Galerkin Method ◽  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Method ◽  
Heterogeneous Media ◽  
Numerical Solutions ◽  
Solution Space ◽  
Convection Diffusion ◽  
Diffusion And Convection ◽  
Convection Dominated ◽  
Diffusion Problems

AbstractIn this paper, we present an efficient computational methodology for diffusion and convection-diffusion problems in highly heterogeneous media as well as convection-dominated diffusion problem. It is well known that the numerical computation for these problems requires a significant amount of computer memory and time. Nevertheless, the solutions to these problems typically contain a coarse component, which is usually the quantity of interest and can be represented with a small number of degrees of freedom. There are many methods that aim at the computation of the coarse component without resolving the full details of the solution. Our proposed method falls into the framework of interior penalty discontinuous Galerkin method, which is proved to be an effective and accurate class of methods for numerical solutions of partial differential equations. A distinctive feature of our method is that the solution space contains two components, namely a coarse space that gives a polynomial approximation to the coarse component in the traditional way and a multiscale space which contains sub-grid structures of the solution and is essential to the computation of the coarse component. In addition, stability of the method is proved. The numerical results indicate that the method can accurately capture the coarse behavior of the solution for problems in highly heterogeneous media as well as boundary and internal layers for convection-dominated problems.

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