Thesis Summary:

This dissertation deals with different aspects of numerical and mathematical analysis of systems of possibly degenerate partial differential equations. Under particular conditions, solutions to these equations in the considered applications exhibit steep gradients, and in the degenerate case, sharp fronts and discontinuities. This calls for a concentration of computational effort in zones of strong variation. To achieve this goal we introduce suitable finite volume methods and fully adaptive multiresolution schemes for spatially one, two and three-dimensional, possibly degenerate reaction-diffusion systems, focusing on sedimentation processes in the mineral industry and traffic flow problems, two and three-dimensional reaction-diffusion systems modelling population dynamics, combustion processes, cardiac propagation and models of pattern formation and chemotaxis in mathematical biology. A novel result is the existence and H¨older regularity of weak solutions of a new nonlinear diffusion model of chemotaxis. Also, for the bidomain equations in electrocardiology, an implicit finite volume method on unstructured meshes is formulated and its convergence to the corresponding weak solution is proved. In order to achieve sparse enough systems while maintaining the same rate of convergence as in the reference methods, the choice of an optimal thresholding strategy for the multiresolution device is addressed. Several numerical experiments confirm the efficiency, good performance and accuracy of the proposed schemes and give insight about the qualitative behavior of the proposed models.

ISI Publications from the Thesis

Raimund BüRGER, Ricardo RUIZ-BAIER: Multiresolution simulation of reaction-diffusion systems with strong degeneracy. Boletin de la Sociedad Espagnola de Matematica Aplicada, vol. 47, pp. 73-80, (2009)

Mostafa BENDAHMANE, Raimund BüRGER, Ricardo RUIZ-BAIER, Jose M. URBANO: On a doubly nonlinear diffusion model of chemotaxis with prevention of overcrowding. Mathematical Methods in the Applied Sciences, vol. 32, 13, pp. 1704-1737. (2009)

Mostafa BENDAHMANE, Raimund BüRGER, Ricardo RUIZ-BAIER, Kai SCHNEIDER: Adaptive multiresolution schemes with local time stepping for two-dimensional degenerate reaction-diffusion systems. Applied Numerical Mathematics, vol. 59, 7, pp. 1668-1692, (2009)

Raimund BüRGER, Ricardo RUIZ-BAIER, Kai SCHNEIDER, Mauricio SEPúLVEDA: Fully adaptive multiresolution schemes for strongly degenerate parabolic equations in one space dimension. ESAIM: Mathematical Modelling and Numerical Analysis, vol 42, 4, pp. 535-563. (2008)

Raimund BüRGER, Ricardo RUIZ-BAIER, Kai SCHNEIDER, Mauricio SEPúLVEDA: Fully adaptive multiresolution schemes for strongly degenerate parabolic equations with discontinuous flux. Journal of Engineering Mathematics, vol. 60, pp. 365-385, (2008)

Other Publications (ISI)

Mostafa BENDAHMANE, Raimund BüRGER, Ricardo RUIZ-BAIER: A multiresolution space-time adaptive scheme for the bidomain model in electrocardiology. Numerical Methods for Partial Differential Equations, vol. 26, 6, pp. 1377-1404, (2010)

Mostafa BENDAHMANE, Raimund BüRGER, Ricardo RUIZ-BAIER: A finite volume scheme for cardiac propagation in media with isotropic conductivities. Mathematics and Computers in Simulation, vol. 80, 9, pp. 1821-1840, (2010)

Raimund BüRGER, Ricardo RUIZ-BAIER, Kai SCHNEIDER: Adaptive multiresolution methods for the simulation of waves in excitable media. Journal of Scientific Computing, vol. 43, 2, pp. 261-290, (2010)