Reservoir Flows

Miscible displacement	One popular technique of enhanced oil recovery is the injection of miscible solvent in the reservoir. Our research on miscible displacement has established results: (i) on convergence of high-order methods in space for smooth solutions; (ii) on convergence of high order methods in space and in time for low regularity solutions; (iii) on robustness of discontinuous Galerkin methods for heterogeneous media, and unstable displacement.
Two-phase flow	Incompressible two-phase flow (such as oil/water) in porous media Our research has established results: (i) on convergence of fully implicit discontinuous Galerkin methods; (ii) on robustness of sequential discontinuous Galerkin methods for heterogeneous media. (iii) on dynamic mesh refinement and coarsening
Three-phase flow	Incompressible three-phase flow in porous media
Black-oil	Molar volume based formulation discretized by discontinuous Galerkin methods
CO2 storage	Fully implicit locally mass conservative schemes

Mathematical Biology

Intestinal edema	Intestinal edema refers to the accumulation of fluid in interstitial spaces of the intestinal wall tissue. The condition causes a decrease in intestinal transit due to decreased intestinal smooth muscle contractility.
Cell motility	Necrotizing enterocolitis is the leading cause of death from gastrointestinal diseases in neonates. We have developed a mathematical model of migration of enterocytes. We have modeled the binding of LPS with TLR4 and the resulting signaling cascade.

Navier-Stokes

Navier-Stokes

Discontinuous Galerkin methods for incompressible Navier-Stokes equations. Our research has established results on: (i) convergence for steady-state and time-dependent problems (ii) splitting schemes (iii) subgrid Eddy viscosity for turbulence models

Numerical Analysis

Coupled methods	Multinumerics schemes combines different discretizations in different parts of the computational domain. Our research has established results on: (i) coupling discontinuous Galerkin methods with finite volume methods (ii) coupling discontinuous Galerkin methods with mixed finite element methods
Uncertainty quantification	Discontinuous Galerkin methods for stochastic PDEs. Our research has established results on: (i) convergence of stochastic discontinuous Galerkin methods for elliptic problems (ii) convergence of Monte Carlo discontinuous Galerkin methods for elliptic problems (iii) uncertainty quantification on coupled flow and transport problems
Low regularity	Convergence of schemes for low regularity solutions
Adaptive meshes	A posteriori error estimates