Eigenvalues for the semi-circulant preconditioning of elliptic operators with the variable coefficients

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:= -Δu + 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 semicirculant preconditioning operator L_cu is constructed by using the leading term of L_vu plus the constant reaction term such that L_cu:= -Δu+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.