Two-Grid Method for Burgers’ Equation by a New Mixed Finite Element Scheme

Abstract

In this article, we present two-grid stable mixed finite element method for the 2D Burgers’ equation approximated by the -P₁ pair which satisfies the inf–sup condition. This method consists in dealing with the nonlinear system on a coarse mesh with width H and the linear system on a fine mesh with width h << H by using Crank–Nicolson time-discretization scheme. Our results show that if we choose H² = h this method can achieve asymptotically optimal approximation. Error estimates are derived in detail. Finally, numerical experiments show the efficiency of our proposed method and justify the theoretical results.

Keywords:

• Burgers’ equation
• two-grid method
• stable conforming finite element
• Crank-Nicolson scheme
• inf-sup condition

AMS Sub ject Classification:

• 65N30
• 65N12
• 65B05
• 35Q30