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Parameter-uniform numerical method for a two-dimensional singularly perturbed convection-reaction-diffusion problem with interior and boundary layers. (English) Zbl 07428978

Summary: We consider a two-dimensional singularly perturbed convection-reaction-diffusion problem that has discontinuities, along lines parallel to x- and y-axes, in the source term, as well as in the convection and reaction coefficients. The coefficient of the highest-order term is a small positive parameter denoted by \(\varepsilon\). Due to the discontinuities, the solution exhibits layers in the interior of the domain, in addition to boundary layers. We propose a decomposition of the solution that yields sharp bounds on its derivatives. A finite difference scheme is constructed on an appropriate Shishkin mesh, and it is established that the computed solution is almost first-order, parameter-uniformly convergent. Numerical results are given to support the theoretical results.

MSC:

65-XX Numerical analysis
35-XX Partial differential equations
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