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

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 imagewhere the nonnegative, bounded coefficient function a(x) of the differential operator may have some isolated zeros in image. 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.