Committee Member

Cox, Christopher; García Vázquez, Concepción; van der Zee, Kristoffer


Dept. of Mathematics


College of Engineering and Computer Science


University of Tennessee at Chattanooga

Place of Publication

Chattanooga (Tenn.)


In this dissertation we focus on the numerical analysis of tumor growth models. Due to the difficulty of developing physically meaningful approximations of such models, we divide the main problem into more simple pieces of work that are addressed in the different chapters. First, in Chapter 2 we present a new upwind discontinuous Galerkin (DG) scheme for the convective Cahn–Hilliard model with degenerate mobility which preserves the pointwise bounds and prevents non-physical spurious oscillations. These ideas are based on a well-suited piecewise constant approximation of convection equations. The proposed numerical scheme is contrasted with other approaches in several numerical experiments. Afterwards, in Chapter 3, we extend the previous ideas to a mass-conservative, positive and energy-dissipative approximation of the Keller–Segel model for chemotaxis. Then we carry out several numerical tests in regimes of chemotactic collapse. These ideas are used later in Chapter 4 to develop a well-suited approximation of two different models related to chemotaxis: a generalization of the classical Keller–Segel model and a model of the neuroblast migration process to the olfactory bulb in rodents’ brains. Now we propose and study a phase-field tumor growth model in Chapter 5. Then, we develop an upwind DG scheme preserving the mass conservation, pointwise bounds and energy stability of the continuous model and we show both the good properties of the approximation and the qualitative behavior of the model in several numerical tests. Next, in Chapter 6, we present two new coupled and decoupled approximations of a Cahn–Hilliard–Navier–Stokes model with variable densities and degenerate mobility that preserve the physical properties of the model. Both approaches are compared in different computational tests including benchmark problems. Consequently, we propose, in Chapter 7, an extension of the previous tumor model including the effects of the surrounding fluid by means of a Cahn–Hilliard–Darcy model for which obtaining a physically meaningful approximation seems rather plausible using the previous ideas. Finally, this and other future lines of research are described, along with the conclusions and the scientific production of the dissertation, in Chapter 8.


I cannot start these words without giving credit to the person who pushed me to start a scientific career and enroll into the doctoral program at the Universidad de Cádiz: J. Rafael Rodríguez Galván. Rafa, as we friendly know him, started as my professor but has soon become a friend – a really good friend – whose support extends far beyond the academic scope. Thank you for being the pillar of my short academic career. In this sense, I also have to mention Francisco Guillén González, Kisko, another close source of support and vast knowledge in mathematics, who never doubted about becoming my PhD advisor once we first contacted him. After these few years of collaboration, the academic relationship that we started has turned into a good friendship that will last many years more. Also, I have to thank the third fundamental person behind this project, Jin Wang, who jumped at the chance of joining this project when the opportunity of starting the joint doctoral program with the University of Tennessee at Chattanooga arose. Thank you for every effort you have made to make this work possible providing the best help throughout the way. Thanks to all the amazing people that have collaborated with me in this project. In particular, Noelia Ortega Román has been really important since the beginning as we joined the research group almost at the same time and we have been working side by side for several years now. In addition, I would like to thank Carmen Castro González and her research group INIBICA INCO-5 for the insightful discussions that arose everytime this group of neuroscientists and our group of mathematicians have met. Finally, I have to mention Giuseppe Viglialoro, Silvia Frassu and Alessandro Columbu, who made Cagliari feel like home and who soon became part of the best friends that academia has put into my life. Then, I would like to thank some other really important people who have tried to make this experience through the doctoral program the least tough they possibly could. In this sense, I must mention the names of Francisco Ortegón Gallego and Juan Carlos García Galindo at the Universidad de Cádiz and Joanne Romagni, Ethan Carver and Christopher L. Cox at the University of Tennessee at Chattanooga. Moreover, although he is not any more in the academia, I will always remember the good conversations with Michael Colvin who always had some really interesting story and some wise piece of advice to tell. Thank you all for somehow becoming part of this project and contribute to its success. Now, I have to give credit to the people who have always been my first and unconditional support, my family, with special mention to my mum Maricarmen, my dad José, my grandparents Uchi and Pepe and my dear partner Laura. They have always been there to hold my hand every time I needed them and to celebrate each of the milestones I have reached on the way. Many times they have told me they are proud of what I have done but I must say, and it is difficult to find the right words, that it is me who has to be proud of them as I have never been on my own – all my little successes are not more than a small part of their own great achievements. Finally, I must talk about my friends, both those who I have made along the process and those who have always been there since the very beginning. I am not going to mention each of their names as I do not want to forget any because, most fortunately, I can say I have dozens of them. Thank you all for the really good moments and the help you have always provided me every time I needed it. I think I will never be able to give you back all the love you have showed to me. All in all, even though it seems that God plays dice in the end, this chaos has somehow guided me to bump into the right people along the way. People who not only made this process enjoyable and fruitful but who are the reason why I am writing these lines today with the biggest smile on my face –and this is far more important than the result itself. Thank you.


Ph. D.; A dissertation submitted to the faculty of the University of Tennessee at Chattanooga in partial fulfillment of the requirements of the degree of Doctor of Philosophy.




Chemotaxis; Computational fluid dynamics; Immunoinformatics; Tumors--Growth


Mass conservation; Pointwise bounds; Energy stability; Diffuse-interface model; Chemotaxis; Multiphase fluid flow


Numerical Analysis and Scientific Computing

Document Type

Doctoral dissertations




xxi, 199 leaves