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# Article

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Keywords:
convection-diffusion problem with dominated convection; Petrov-Galerkin method; reaction-diffusion equation; test functions; Petrov-Galerkin method; Dirichlet problem; algorithm; numerical examples
Summary:
A general construction of test functions in the Petrov-Galerkin method is described. Using this construction; algorithms for an approximate solution of the Dirichlet problem for the differential equation $-\epsilon u^n + pu' + qu=f$ are presented and analyzed theoretically. The positive number $\epsilon$ is supposed to be much less than the discretization step and the values of $\left|p\right|,q$. An algorithm for the corresponding two-dimensional problem is also suggested and results of numerical tests are introduced.
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