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 are especially interested in space-time Discontinuous Galerkin (DG) methods jcp2013 and space-time Hybridizable Discontinuous Galerkin (HDG) methods jcp2012, dirk2013. Recently we also introduced a hybridizable discontinuous Galerkin method for the Stokes equations siam2017 and the Navier-Stokes equations jsc2018 for which the approximate velocity field is pointwise divergence-free and H(div)-conforming. We introduced preconditioners for these methods in arXiv2018.

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 to be able to do large scale, three dimensional computations. Some preliminary results of mantle and magma velocity fields in a 3D subduction zone are given below. See also our article on 2-field preconditioners for coupled magma/mantle dynamics on siam2014, a new 3-field preconditioning approach on siam2015, as well as our articles comparing numerical simulation with experimental data on jgr2014 (2D) and g3-2015 (3D). Supporting material (codes) for the above papers are available on cambridge1, cambridge2 and bitbucket. See the FoaLab website for more information on this project.

We developed an hp-Multigrid as Smoother algorithm in which (semi-coarsening) h-multigrid acts as smoother for p-multigrid. Excellent convergence rates are achieved for low and high cell Reynolds numbers and highly stretched meshes. See the articles jcp2012a and jcp2012b and the extended report twente2011.

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 jcp2008 presents a general framework for discontinuous Galerkin discretizations of these nonconservative hyperbolic equations. We applied this framework to shallow two-phase flows in cmame2009 and applications for river bed evolution in cmame2008.