The paper is devoted to numerical solution of the stationary \((\partial \varphi/ \partial t=0)\) and nonstationary convection-diffusion equations \({\partial \varphi \over \partial t} -\varepsilon \text{div} (\mu \nabla\varphi) +\text{div} (\vec u\varphi) +(1+\varepsilon) a\varphi = f\), \((x,y,t) \in \Omega \times (0,T)\), for sufficiently small viscosity coefficient \(\varepsilon\). The authors prove the convergence of solutions \(\varphi\) for both problems, as \(\varepsilon \to 0\), to the corresponding solutions \(\varphi_0\) for nonperturbed problems with \(\varepsilon =0\). The special attention is paid to the consistency of boundary conditions in the original and the limiting problems. For the numerical approximation, the authors use the finite element method with respect to space variables and an implicit first-order finite difference scheme with respect to the time variable. As an example, the authors solve numerically the problem of the impurity propagation in the Cagliari Bay.