Thesis Summary:

Watercourses, in addition to being an essential resource for the life of man, are used to remove wastes. With the growing population and number of factories that throw their wastes into them today, a great number of rivers have been contaminated. To prevent this situation, it is necessary to develop models of water quality that allow predicting the concentration of pollutants in the event of possible scenarios of discharge and discharge of contaminants. In order to model the transport of pollutants in the river, in this thesis we have considered the advection-reaction-diffusion equation, where the discharge is modeled with a regular source if it is diffuse, and with a Dirac delta source if it is punctual. To solve this equation efficiently we use an adaptive scheme, based on a finite element stabilized method combined with a posteriori error estimators. In the first part of the thesis, we consider diffuse discharges. Here we introduce an adaptive finite element scheme for the transport equation, considering sources in L2 (). We have developed error estimators a posteriori of both residual type and based on local problem solving. Both estimators allow to obtain correctly refined meshes. Then the Laplace equation is studied with a delta source supported at an interior point of the domain. It is shown that the solution of this problem belongs to W1, p (), 1p <2, and therefore to Lr (), r <. For this reason, some residual error-type error-equivalent error estimators are introduced in both norm W1, p () and Lr (). Finally, the equation of transport with delta sources is solved by means of a stabilized adaptive scheme, for which a posteriori residual type estimators equivalent to the error are developed, which allow to obtain correctly refined meshes. This makes it possible to obtain good approximations of the concentration of pollutants, which is very useful when analyzing possible discharges in rivers. In all the cases an abundant numerical experimentation is presented, which allows us to establish the good behavior of the developed estimators.

ISI Publications from the Thesis

Rodolfo ARAYA, Edwin BEHRENS, Rodolfo RODRíGUEZ: Error estimators for advection-reaction-diffusion equations based on the solution of local problems. Journal of Computational and Applied Mathematics, vol. 206, pp. 440-453, (2007)

Rodolfo ARAYA, Edwin BEHRENS, Rodolfo RODRíGUEZ: An adaptive stabilized finite element scheme for a water quality model. Computer Methods in Applied Mechanics and Engineering, vol. 196, 29-30, pp. 2800-2812, (2007)

Rodolfo ARAYA, Edwin BEHRENS, Rodolfo RODRíGUEZ: A posteriori error estimates for elliptic problems with Dirac delta source term. Numerische Mathematik, vol. 105, 2, pp. 193-216, (2006)

Rodolfo ARAYA, Edwin BEHRENS, Rodolfo RODRíGUEZ: An adaptive stabilized finite element scheme for the advection-reaction-diffusion equation. Applied Numerical Mathematics, vol. 54, pp. 491-503, (2005)