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 {An (a)} coming from the discretization, using centered finite differences of minimal order, of elliptic (or semielliptic) differential operators L (a, u) of the form (1)fenced((- frac(d, d x) fenced(a (x) frac(d, d x) u (x)) = f (x) on Ω = (0, 1),; Dirichlet B.C. on ∂ Ω,))where the nonnegative, bounded coefficient function a (x) of the differential operator may have some isolated zeros in over(Ω, -) = Ω ∪ ∂ Ω. More precisely, we state and prove the explicit form of the inverse of {An (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, and then we present various numerical experiments, showing that similar results must hold true in 2D as well.