High-order matrix-free solvers
Much of my work is focused on the development and analysis of matrix-free preconditioners for high-order finite element discretizations. The linear systems associated with high-order discretizations are often ill-conditioned, and efficient sum-factorized implementations are necessarily matrix-free, requiring the construction of preconditioners without access to the matrix entries.
Much of my work on this subject has been focused on low-order refined preconditioning (LOR), also known as FEM-SEM preconditioning. These methods construct spectrally equivalent lowest-order discretizations on refined meshes. Algebraic or semi-structured multigrid methods constructed with the low-order discretization can be used to effectively precondition the original high-order system.
Recent advanced on LOR preconditioning include:
- Degree-robust preconditioners on triangular meshes (2026)
- Solvers for pressure-robust interior penalty Stokes discretizations (2025)
- Fast GPU-accelerated saddle-point solvers (2024)
- Full end-to-end GPU acceleration (2023)
- Comprehensive theory for LOR on the de Rham complex (2023)
- Uniform solvers for DG methods with hp refinement (2023)
Pictured: high-order magnetic field streamlines using elements (left), hexahedral mesh (right).
Open-source mathematical software
I am a core contributor to the MFEM finite element library, whose repository can be found on GitHub. I also contributed the Julia interface to libCEED, available as LibCEED.jl.
These libraries provide access to cutting-edge discretizations and solvers, enabling reproducible computational science. Both libraries support extensive matrix-free capabilities and GPU acceleration.
Discontinuous Galerkin methods
Invariant domain preserving methods. In our 2021 CMAME paper, we developed provably robust invariant-domain-preserving high-order DG methods by using convex limiting to blend a high-order DG method with a compatible sparsified low-order method. In subsequent work, we showed that the sparsified low-order method is equivalent to a certain finite volume method, and considered general combinations or high-order target schemes, robust low-order methods, and subcell blending approaches.
Entropy-stable line-based sparse discretizations. In our 2019 JSC paper, we analyzed the sparse line-DG method and introduced a modification that enforces discrete entropy stability for hyperbolic conservation laws.
Pictured: Riemann problem for Euler equations (left), and entropy stability and conservation properties (right).
Kronecker-product preconditioners for extremely high-order DG methods. We developed tensor-product preconditioners that use sum-factorization to reduce the cost of implicit solvers for very high-order DG methods. These methods were applied to LES and DNS flow problems in 2D and 3D. Our paper was awarded first-place in the 2017 AIAA CFD student paper competition.
Algorithmic details and numerical results appear in our 2018 JCP paper, with an extension to second-order systems in an interior penalty formulation.
Pictured: Taylor-Green vortex (left), and degree-30 inviscid flow over a bump (right).
Fully-implicit Runge-Kutta time integration for DG. For Radau-IIA fully-implicit Runge-Kutta methods, we transformed the coupled DG systems into a sparser formulation that can be preconditioned efficiently.
We used a stage-parallel shifted ILU(0) preconditioner; details appear in our 2017 JCP paper.
The same approach was extended to adjoint equations for PDE-constrained optimization and applied to transonic buffeting at high Reynolds number.
Pictured: Stage-parallel strategy for hybrid shared-distributed memory supercomputers (left), and LES over a NACA in 3D using 3-stage fifth-order IRK.
Convergence of iterative solvers for polygonal DG methods. We studied iterative solvers for discontinuous Galerkin methods on arbitrary polygonal meshes. A von Neumann analysis and numerical experiments showed that polygonal meshes can lead to faster iterative convergence. More details appear in our CAMCoS paper.
Pictured: a polygonal mesh refined in the vicinity of a supersonic shock.
High-order time integration and low-Mach combustion
This research project, in collaboration with scientists from Center for Computational Science and Engineering at Lawrence Berkeley National Lab, was concerned with the development of high-order time integration schemes for low-Mach number combustion. A new strategy for employing multi-implicit spectral deferred corrections (MISDC) was used in conjunction with a fourth-order finite volume method in order to obtain overall fourth-order convergence for hydrogen, methane, and dimethyl ether flame simulations. The results were published in our 2016 paper.
This line of work was later extended to partitioned multiphysics problems in our 2020 JCP paper, which developed arbitrarily high-order partitioned SDC methods using pre-existing single-physics solvers and studied their stability and accuracy on several multiphysics model problems.