Dimension splitting algorithm for a three-dimensional elliptic equation

Abstract

This paper presents a finite-element dimension splitting algorithm (DSA) for a three-dimensional (3D) elliptic equation in a cubic domain. The main idea of DSA is that a 3D elliptic equation can be transformed into a series of two-dimensional (2D) elliptic equations in the X–Y plane along the Z-direction. The convergence speed of the DSA for a 3D elliptic equation depends mainly on the mesh scale of the Z-direction. P ₂ finite-element discretization in the Z-direction for DSA is adopted to accelerate the convergence speed of DSA. The error estimates are given for DSA applying P ₁ or P ₂ finite-element discretization in the Z-direction. Finally, some numerical examples are presented. We apply the parallel solving technology to our numerical examples and obtain good parallel efficiency. These numerical experiments test and verify theoretical results.

Keywords:

• dimension splitting algorithm
• elliptic equation
• finite-element method
• parallel solving technology
• error estimate

2000 AMS Subject Classifications :

• 65Y05
• 65N30

Acknowledgements

This work was supported by the National Natural Science Foundation of China (11171269, 10871156) and Jiaoda Foundation (2009xjtujc30). Especially, we thank the anonymous referees very much for their helpful comments and suggestions, which led to substantial improvements of the presentation.