Asymptotic results on the condition number of FD matrices approximating semi-elliptic PDEs

Abstract

This work studies the asymptotic behavior of the spectral condition number of the matrices $A_{nn}$ arising from the discretization of semi-elliptic partial differential equations of the form \bdm -\left( a(x,y)u_{xx}+b(x,y)u_{yy}\right)=f(x,y), \edm on the square $\Omega=(0,1)^2,$ with Dirichlet boundary conditions, where the smooth enough variable coefficients $a(x,y), b(x,y)$ are nonnegative functions on $\overline{\Omega}$ with zeros. In the case of coefficient functions with a single and common zero, it is discovered that apart from the minimum order of the zero also the direction that it occurs is of great importance for the characterization of the growth of the condition number of $A_{nn}$. On the contrary, when the coefficient functions have non intersecting zeros, it is proved that independently of the order their zeros, and their positions, the condition number of $A_{nn}$ behaves asymptotically exactly as in the case of strictly elliptic differential equations, i.e., it grows asymptotically as $n^2$. Finally, the more complicated case of coefficient functions having curves of roots is considered, and conjectures for future work are given. In conclusion, several experiments are presented that numerically confirm the developed theoretical analysis.