Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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EIGENVALUES FOR THE SEMI-CIRCULANT PRECONDITIONING OF ELLIPTIC OPERATORS WITH THE VARIABLE COEFFICIENTS

  • Kim, Hoi-Sub (DEPARTMENT OF MATHEMATICS AND INFORMATION KYUNGWON UNIVERSITY) ;
  • Kim, Sang-Dong (DEPARTMENT OF MATHEMATICS KYUNGPOOK NATIONAL UNIVERSITY) ;
  • Lee, Yong-Hun (DIVISION OF MATHEMATICS AND STATISTICAL INFORMATICS(INSTITUTE OF APPLIED STATISTICS) CHONBUK NATIONAL UNIVERSITY)
  • Published : 2007.05.31
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Abstract

We investigate the eigenvalues of the semi-circulant preconditioned matrix for the finite difference scheme corresponding to the second-order elliptic operator with the variable coefficients given by $L_vu\;:=-{\Delta}u+a(x,\;y)u_x+b(x,\;y)u_y+d(x,\;y)u$, where a and b are continuously differentiable functions and d is a positive bounded function. The semi-circulant preconditioning operator $L_cu$ is constructed by using the leading term of $L_vu$ plus the constant reaction term such that $L_cu\;:=-{\Delta}u+d_cu$. Using the field of values arguments, we show that the eigenvalues of the preconditioned matrix are clustered at some number. Some numerical evidences are also provided.

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References

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