Grid Approximation of a Singularly Perturbed Parabolic Equation with Degenerating Convective Term and Discontinuous Right-hand Side

The grid approximation of an initial-boundary value problem is considered for a singularly perturbed parabolic convection-diffusion equation with a convective flux directed from the lateral boundary inside the domain in the case when the convective flux degenerates inside the domain and the right-hand side has the first kind discontinuity on the degeneration line. The high-order derivative in the equation is multiplied by ε2, where ε is the perturbation parameter, ε ∈ (0, 1]. For small values of ε, an interior layer appears in a neighbourhood of the set where the right-hand side has the discontinuity. A finite difference scheme based on the standard monotone approximation of the differential equation in the case of uniform grids converges only under the condition N−1 = o(ε), N−1 0 = o(1), where N +1 and N0 +1 are the numbers of nodes in the space and time meshes, respectively. A finite difference scheme is constructed on a piecewise-uniform grid condensing in a neighbourhood of the interior layer. The solution of this scheme converges εuniformly at the rate O(N−1 lnN +N−1 0 ). Numerical experiments confirm the theoretical results.