<abstract><p>In this paper we describe a coupled model for flow and microbial growth as well as nutrient utilization. These processes occur within and outside the biofilm phase formed by the microbes. The primary challenge is to address the volume constraint of maximum cell density but also to allow some microbial presence outside the contiguous biofilm phase. Our model derives from the continuum analogues of the mechanism of cell shoving introduced in discrete biomass models, and in particular from the models exploiting singular diffusivity as well as from models of variational inequality type which impose explicit constraints. We blend these approaches and propose a new idea to adapt the magnitude of the diffusivity automatically so as to ensure the volume constraint without affecting the reactions; this construction can be implemented in many variants without deteriorating the overall efficiency. The second challenge is to account for the flow and transport in the bulk fluid phase adjacent to the biofilm phase. We use the Brinkman flow model with a spatially variable permeability depending on biomass amount. The fluid flow allows some advection of the nutrient within the biofilm phase as well as for the flow even when the pores are close to being plugged up. Our entire model is monolithic and computationally robust even in complex pore-scale geometries, and extends to multiple species. We provide illustrations of our model and of related approaches. The results of the model can be easily post—processed to provide Darcy scale properties of the porous medium, e.g., one can predict how the permeability changes depending on the biomass growth in many realistic scenarios.</p></abstract>

<abstract><p>We consider heat conduction models with phase change in heterogeneous materials. We are motivated by important applications including heat conduction in permafrost, phase change materials (PCM), and human tissue. We focus on the mathematical and computational challenges associated with the nonlinear and discontinuous character of constitutive relationships related to the presence of free boundaries and material interfaces. We propose a monolithic discretization framework based on lowest order mixed finite elements on rectangular grids well known for its conservative properties. We implement this scheme which we call P0-P0 as cell centered finite differences, and combine with a fully implicit time stepping scheme. We show that our algorithm is robust and compares well to piecewise linear approaches. While various basic theoretical properties of the algorithms are well known, we prove several results for the new heterogeneous framework, and point out challenges and open questions; these include the approximability of fluxes by piecewise continuous linears, while the true flux features a jump. We simulate a variety of scenarios of interest.</p></abstract>