Convergence rate estimates for difference schemes of one class of divergent equations (Q1116665)

We derive some rate of convergence estimates for difference schemes of the quasilinear equation \[ -d/dx[x^{\alpha}k(x,u,du/dx)]+k_ 0(x,u,du/dx)=f(x),\quad 0<x<1; \] \[ u(1)=0,\quad \lim_{x\to 0}x^{\alpha}k(x,u,du/dx)=0 \] which diverge for \(x=0\). The exact solution of the original problem is assumed to go to infinity with rate \(O(1/x^{\tau})\). It is shown that if \(\gamma\leq | \alpha /2-1/2- \epsilon |\), then the difference solution converges to the exact solution of the differential problem in the difference norm with rate \(O(h^{1/2}+h^{\epsilon /2})\).