Multigrid method with a new interpolation operator

Abstract

We introduce new techniques to design interpolation in multigrid methods for elliptic problems with discontinuous coefficients. The new techniques employ the Nelder–Mead simplex algorithm and skills in space geometry. The Nelder–Mead algorithm was used to minimize a scalar-valued function, which is a sum of distances from a point to four planes. We derived interpolation scheme in space geometry. We observed that new interpolation is better than traditional bilinear interpolation and cubic interpolation, as prolongation operator in multigrid methods.

Keywords:

• multigrid
• discontinuous coefficients
• interpolation
• Nelder–Mead algorithm

1991 AMS Subject Classifications :

• 65N55
• 65N30
• 65N06

Acknowledgements

The author would like to thank Professor Yunqing Huang for his encouragement and many discussions and also the referees for several constructive comments.