Resumen de Asymptotic approximations for singularly perturbed convection diffusion problems with discontinuous boundary data
This thesis is focused on the study of two and three dimensional singularly-perturbed convection-diffusion problems of parabolic and elliptic type defined over different unbounded and bounded domains that present, besides the small perturbation parameter, another source of singular behaviour for the solution:
discontinuities of the boundary/initial data or of their derivatives, The difficulty originated by discontinuities of the Dirichlet data is a subject of recent interest because there are not many theoretical nor numerical results about this problem. In this thesis, we analyze the effect that jump discontinuities of the boundary/initial conditions or of their derivatives has on the singular behaviour of the solution. For this purpose we consider several examples of parabolic or elliptic convection-diffusion singularly perturbed problems with constant coefficients defined in bounded and unbounded domains in 2 or 3 dimensions with discontinuous data (in the Dirichlet conditions or in its derivatives). For every one of the examples analyzed, we obtain an exact representation of the solution in terms of a Laplace or Fourier integral or in terms of a series of Fourier integrals. Then, using the classical asymptotic method for integrals which contain saddle points near a pole of the integrand, we construct, for every analyzed problem, an asymptotic expansion of the solution in the singular limit in which the perturbation parameter goes to 0. The first term of this expansion contains always an error function (in three dimensions it contains a product of error functions). These expansions are not valid near the discontinuities of the Dirichlet data. We use distributional methods for asymptotic expansions of integrals to derive convergent expansions of the solution of every problem near de discontinuities in terms of powers of the distance to the discontinuity.
Most of the analyzed examples show that the first term of this second expansion is a
