A Petrov-Galerkin approximation of convection-diffusion and reaction- diffusion problems (Q1184930)

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.