Numerical solution of a singularly perturbed elliptic-hyperbolic partial differential equation on a nonuniform discretization mesh (Q1210173)

The authors consider the upwind difference scheme for the following singular perturbation problem \[ \varepsilon(u_{xx}+u_{yy})+a(x,y)u_ x+b(x,y)u_ y-c(x,y)u=f(xy) \] \((x,y)\in \Omega=[0,1]\times [0,1]\) with the boundary condition \(u\mid_{\partial\Omega}=0\). On a special discretization mesh it is proved that the solution of the upwind difference scheme is first order convergent, uniformly in the small parameter \(\varepsilon\), to the solution of our problem. Numerical results are finally provided.