- Reduction-based Algebraic Multigrid for Upwind Discretizations (jointly with Tom Manteuffel, Steve McCormick, John Ruge and Ben Southworth), submitted to SIAM J. Sci. Comput., ArXiv
- A Comparison of Finite Element Spaces for H(div) Conforming First-Order-System Least Squares (jointly with Chris Leibs and Tom Manteuffel), submitted to SIAM J. Sci. Comput.
- First-Order System Least Squares for generalized-Newtonian Coupled Stokes-Darcy Flow, Numer. Methods Partial Differential Eq., 31(4), 1150-1173, 2015
One of the main developers of LEAP (Least-Squares Package), which provides an easy-to-use lightweight python-based frontend for the Least-Squares FEM using FEniCS. To this point LEAP can be applied to (time-dependent) (non)linear PDEs, as long as they are provided as a first-order formulation. Solution techniques (and arbitrary combinations of these) include
- Time-stepping schemes
- Nested Iteration (adaptive refinement based on the Least-Squares functional) with various refinement techniques
- Nonlinear solvers (with a user provided linearization scheme), with various line-search methods and possibly nonlinear boundary conditions
- An easy-to-use interface to iterative solvers via the PETsC interface (for example hypre or PyAMG), with easy access to many parameters
- Use of mixed finite element spaces, as long as they are supported in FEniCS
- Adjoint methods, such as FOSLL* or Hybrid Methods for linear PDEs
For example the LSFEM (using arbitray FE-spaces provided in FEniCS) solution of a time-dependent nonlinear system of PDEs (with possibly nonlinear boundary conditions) using adaptive refinement strategies for every time-step and solving the linear systems by an AMG method, can be easily done by just providing the PDE itself, the boundary conditions and the linearization. The implementation is done such that above functionalities do work in parallel (using mpi).
LEAP is not yet open for public, but can be used by invitation to the bitbucket library.