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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    ∆    , where a and b are continuously differentiable functions and d is a positive bounded function. The semi-circulant preconditioning operator  ∆. 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.

EIGENVALUES FOR THE SEMI-CIRCULANT PRECONDITIONING OF ELLIPTIC OPERATORS WITH THE VARIABLE COEFFICIENTS