A Newton-like method for solving a non-smooth elliptic equation

Abstract

In this paper, we use finite element method to discrete a non-smooth elliptic equation and present some error estimates. Non-smooth Newton-like method is applied to solve the discrete problem. Since Newton's equations have a very bad conditioner when the mesh-size is finer, multigrid technique is used to solve the subproblems. It is shown that if we use V-cycle or cascadic multigrid as an inner iterator, an (nearly) optimal property can be obtained. Numerical results are illustrated to confirm the error estimates we obtained and the efficiency of the non-smooth Newton-like method combining with multigrid technique. Especially, if the mesh-size h becomes much smaller, the method can save substantial computational work.

Keywords:

• non-smooth equation
• finite element method
• non-smooth Newton method
• multigrid technique

AMS Subject Classificaton :

• 49M15
• 65N30
• 65N55

Acknowledgements

This work was supported by the NNSF of China grant 10971058.

Additional information

Notes on contributors

Haixiong Yu

Haixiong Yu is now affiliated with the Department of Science, Nanching Institute of Technology, Nanchang, China