iFEM is a MATLAB software package containing robust, efficient, and easy-following codes for the main building blocks of adaptive finite element methods on unstructured simplicial grids in both two and three dimensions. Besides the simplicity and readability, sparse matrixlization, an innovative programming style for MATLAB, is introduced to improve the efficiency. In this novel coding style, the sparse matrix and its operations are used extensively in the data structure and algorithms.

We study fine details of spreading of reactive processes in multidimensional
inhomogeneous media. In the real world, one often observes a transition from one equilibrium (such as unburned areas in forest fires) to another (burned areas)to happen over short spatial as well as temporal distances. We demonstrate that this phenomenon also occurs in one of the simplest models of reactive processes, reaction-diffusion equations with ignition reaction functions, under very general hypotheses.

Specifically, in up to three spatial dimensions, the width (both in space and time) of the zone where the reaction occurs turns out to remain uniformly bounded in time for fairly general classes of initial data. This bound even becomes independent of the initial data and of the reaction function after an initial time interval. Such results have recently been obtained in one dimension, in which one can even completely characterize the long term dynamics of general solutions to the equation, but are new in dimensions two and three. An indication of the added difficulties is the fact that three dimensions turns out to indeed be the borderline case, as the bounded-width result is in fact false for general inhomogeneous media in four and more dimensions.

More and more, traveling waves are observed inside individual cells. These waves can be pulses of biochemical factors (diffusing proteins or metabolites) but also mechanical factors (such as the cell cortex). One example of mechanical traveling wave is offered by cellular blebs, pressure-driven “bubbles” on the cell surface implicated in cell division, apoptosis and cell motility. Blebs exhibit a range of behaviors including contracting in place, travel around the cell’s periphery, or repeated blebbing, making them biophysically interesting. Mechanical traveling waves are naturally modeled using “non-local” integro-PDEs, which lack the theoretical tools available for reaction-diffusion waves. This lack obfuscates simple questions such as what determines if a bleb will travel or not, and, if it travels, what determines its velocity? We present results in two parts: First, we develop a simple model of the cell surface describing the membrane, cortex, and adhesions, including the slow timescale cortical healing (treating implicitly the fast timescale of fluid motion). We find traveling and stationary blebs, which we characterize through numerical simulation. In the second part, we review the so-called Maxwell condition for reaction-diffusion systems that determines whether an excitation will travel or recover in place. We present our progress in deriving an analogue of the Maxwell condition for non-local integro-PDEs suitable for our cell surface model. This condition allows the theoretical (simulation-free) elucidation of blebbing including bleb travel. 

A second order energy stable numerical scheme is presented for the two and three dimensional Cahn-Hilliard equation, with Fourier pseudo-spectral approximation in space. The convex splitting nature assures its unique solvability and unconditional energy stability. Meanwhile, the implicit treatment of the nonlinear term makes a direct nonlinear solver not available, due to the global nature of the pseudo-spectral spatial discretization. In turn, a linear iteration algorithm is proposed 
to overcome this difficulty, in which a Douglas-Dupont-type regularization term is introduced. As a consequence, the numerical efficiency has been greatly improved, since the highly nonlinear system can be decomposed as an iteration of purely linear solvers. Moreover, a careful nonlinear analysis shows a contraction mapping property of this linear iteration, In addition, a maximum norm bound of numerical solution is also derived at a theoretical level. A few numerical examples 
are also presented in this talk.