Part I: A Virtual Node Method for Elliptic Interface Problems. Part II: Local and Global Theory of Aggregation Equations with Nonlinear Diffusion

In Part I, the author presents an accurate and efficient method for solving elliptic interface problems or elliptic problems in irregular domains. Such problems occur in a wide variety of applications in physics and engineering and are regarded as computationally difficult, particularly when the interface or boundaries are moving. The work is in collaboration with James H. von Brecht, Siwei Zhu, Eftychios Sifakis and Joseph Teran and appears also in the publication [19]. We introduce a second order virtual node method for approximating elliptic interface problems on a uniform Cartesian grid. The use of a regular Cartesian grid simplifies the implementation and permits straightforward Lagrange multiplier spaces while re-airing second order accuracy in L infinity in numerical experiments. Our approach uses duplicated Cartesian bilinear elements along the interface to introduce additional "virtual" nodes that accurately account for the lack of regularity across the surface.;Part II discusses the work undertaken by the author and his collaborators Nancy Rodriguez and Andrea Bertozzi on the class of aggregation equations with nonlinear diffusion, which represent a generalization of the classical parabolic-elliptic Patlak-Keller-Segel system for chemotaxis. These models represent the competition between nonlocal self-attraction and diffusion. Local theory, such as existence, uniqueness and continuation is first discussed. A suitable notion of L1-criticality is introduced for inhomogeneous problems and the sharp critical mass is identified: uniform bounds Linfinity are derived for solutions to subcritical problems and critical problems with less than critical mass and finite time blowup is derived for a class of supercritical problems and critical problems with larger than critical mass. Global, dissipating solutions are constructed under certain, reasonably general hypotheses and the asymptotic profiles are shown to agree with the self-similar Barenblatt solutions to the homogeneous diffusion equations.