The necessary and sufficient condition of uniformly convergent difference schemes for the elliptic-parabolic partial differential equation with a small parameter
A homogeneous Dirichlet problem for the sigularly perturbed equation \[ \partial^ 2u/\partial x^ 2+\epsilon \partial^ 2u/\partial y^ 2- a(x,y)\partial u/\partial y=f(x,y),\quad \epsilon >0,\quad a(x,y)>0 \] is considered. When \(\epsilon =0\) this problem reduces to the corresponding boundary value problem for a parabolic equation. The authors seek for a finite difference scheme with the properties: (i) it adapts to the behavior of the boundary layer; (ii) when \(\epsilon\) \(\to 0\) the approximate solution is consistent with the exact solution; (iii) when the step size of the mesh h tends to zero the approximate solution converges uniformly in \(\epsilon\) to the exact solution. The authors construct a difference scheme and find a necessary condition for (iii). Under the assumption that a(x,y) is a constant and f(x,y) is smooth enough this condition is proved to be sufficient for (iii) and the rate of convergence is O(h).
scientific article; zbMATH DE number 3915562
scientific article