Convergence of the Difference Solutions of a Dirichlet Problem With a Discontinuous Derivative of the Boundary Function for a Singularly Perturbed Convection-Diffusion Equationстатья
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Аннотация:Abstract. We consider a Dirichlet problem for a singularly perturbed convection-diffusion equation
with constant coefficients in a rectangular domain in the case when the convection is parallel to the
horizontal faces of the rectangular and directed to the right while the first derivative of the boundary
function is discontinuous on the left face. Under these conditions the solution of the problem has a regular
boundary layer in the neighborhood of the right face, two characteristic boundary layers near the top
and bottom faces, and a horizontal interior layer due to the non-smoothness of the boundary function.
We show that on the piecewise uniform Shishkin meshes refined near the regular and characteristic
layers, the solution given by the classical five-point upwind difference scheme converges uniformly to the
solution of the original problem with almost first-order rate in the discrete maximum norm. This is the
same rate as in the case of a smooth boundary function. The numerical results presented support the
theoretical estimate. They show also that in the case of the problem with a dominating interior layer
the piecewise uniform Shishkin mesh refined near the layer decreases the error and gives the first-order
convergence.