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)}{An(a)} coming from the discretization, using centered finite differences of minimal order, of elliptic (or semielliptic) differential operators L(a,u)L(a,u) of the form equation(1)-ddxa(x)ddxu(x)=f(x)onΩ=(0,1),Dirichlet B.C. on∂Ω,where the nonnegative, bounded coefficient function a(x)a(x) of the differential operator may have some isolated zeros in Ω¯=Ω∪∂Ω. More precisely, we state and prove the explicit form of the inverse of {An(a)}{An(a)} and some formulas concerning the relations between the orders of zeros of a(x)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.