Divergence conforming hybridizable and embedded discontinuous Galerkin methods
In RW17 and RW18a, we introduced a class of very simple divergence-conforming hybridizable discontinuous Galerkin (HDG) methods for the Stokes and the Navier-Stokes equations. These methods result in an approximate velocity field that is pointwise divergence-free. As a result, this method conserves mass exactly, is locally momentum conserving, energy stable, and pressure robust. We introduced also a new mixed embedded-hybridized discontinuous Galerkin (EDG-HDG) method in RW20. This new method has the same properties as the HDG method, but has fewer global degrees-of-freedom and the resulting linear system is better suited for fast iterative solvers RW18b, RW21.
Space-time finite element methods for PDEs on deforming domains
For many flows of fluids the boundary containing the fluid changes shape. Examples include the ocean, where the interface between water, air and land is constantly evolving; and rotating wind-turbines in strong enough wind flow. We are developing higher-order accurate numerical methods for solving partial differential equations on time-dependent domains. We introduced a space-time HDG method for the Navier-Stokes probem in RC12. More recently, by extending RW18a and RW20 to the space-time setting, we introduced, respectively, the space-time HDG HR19 and space-time EDG-HDG HR20 methods that result in an exactly divergence-free and divergence-conforming discrete velocity field, even on time-dependent domains with moving meshes.
Computational tools for magma dynamics of subduction zones: finite element models and efficient solvers
The goal of this project is to understand more about how magma is formed and how it is transported in subduction zones. For this we are developing efficient solvers/preconditioners. See the FoaLab website for more information on this project.
Discontinuous Galerkin finite element methods for nonconservative hyperbolic equations
Many models describing flows contain nonconservative products. These include the shallow water equations coupled to sediment continuity equations and compressible two-phase flow equations. Hyperbolic pdes with non-conservative products cannot be written in divergence form. Our work in RBV08 presents a general framework for discontinuous Galerkin discretizations of these nonconservative hyperbolic equations.