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A Petrov-Galerkin approximation of convection-diffusion and reaction- diffusion problems. (English) Zbl 0748.65061
A new construction of test functions in the Petrov-Galerkin method is described. Using this construction, the author proposes algorithms for solving the Dirichlet problem for the differential equation $$-\varepsilon u''+pu'+qu=f$$, where the positive number $$\varepsilon$$ is supposed to be much less than the discretization step and the values of $$| p|$$, $$q$$.
An algorithm for the corresponding two-dimensional problem is also suggested and numerical examples are given to demonstrate the effectiveness of the proposed algorithms.
Reviewer: K.Najzar (Praha)

##### MSC:
 65L10 Numerical solution of boundary value problems involving ordinary differential equations 65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 35J25 Boundary value problems for second-order elliptic equations 34B05 Linear boundary value problems for ordinary differential equations 34E15 Singular perturbations, general theory for ordinary differential equations
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