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Computational Method for Convection-Dominated Problem

In this paper convection-dominated problem is considered. The problem will lead to large oscillation since the coefficient of diffusion term is small. Firstly, the analytical solution is decomposed into the smooth component and the singular component. Secondly, Runge-Kutta method is applied to solve the equation outside the boundary layer. Thirdly, Petrov-Galerkin finite element method with piecewise-exponential test function and the piecewise-linear trial function is constructed in order to solve the boundary layer. Finally, the error estimation is given.