The conditioning of FD matrix sequences coming from semi-elliptic differential equations

In this paper we are concerned with the study of spectral properties of the sequence of matrices {A(n)(a)} coming from the discretization, using centered finite differences of minimal order, of elliptic (or semielliptic) differential operators L(a, u) of the form {-d/dx (a(x)d/dx u(x)) = f(x) on Omega = (0, 1), Dirichlet B.C. on theta Omega, (1) where the nonnegative, bounded coefficient function a(x) of the differential operator may have some isolated zeros in U (Omega) over bar = Omega boolean OR theta Omega. More precisely, we state and prove the explicit form of the inverse of {A(n)(a)} and some formulas concerning the relations between the orders of zeros of a(x) and the asymptotic behavior of the minimal eigenvalue (condition number) of the related matrices. As a conclusion, and in connection with our theoretical findings, first we extend the analysis to higher order (semi-elliptic) differential operators,