Research

I am interested in the development and analysis of numerical schemes for PDEs and the corresponding linear solvers. Most recently, I have been working on methods arising from the Hodge decomposition for divergence-free fields to tackle electromagnetics problems on general topology domains. I am also interested in applications where PDE constraints should be satisfied exactly in the discretization, for example, divergence-free conditions in electromagnetics problems. More generally, I have focused on finite element and finite difference methods that arise from finite-element exterior calculus and discrete exterior calculus, respectively. I also work on developing robust linear solvers and multigrid methods for these problems.

Here is my CV.

Finite element methods based on Hodge decomposition

Many standard numerical methods face challenges on complicated geometries with holes due to the introduction of harmonic forms in the solution space. To overcome this, one can use the Hodge decomposition, which explicitly accounts for the harmonic forms. In the case of high-order PDEs, like the quad-curl problem, this also allows for the fourth order problem to be reformulated in terms of standard second order problems.

Convection-dominated diffusion problems

An essential result from PDEs regarding scalar elliptic equations is the maximum principle. However, in discretizations of elliptic equations, loss of a discrete maximum principle results in numerical solutions polluted by oscillations and artifacts in the convection-dominated regime. The development of monotone schemes, schemes which obey a discrete maximum principle, are essential for both the scalar and vector convection-dominated equations to guarantee high-fidelity and physical soutions.

Mimetic finite differences and multigrid for Maxwell's equations

Maxwell's equations are the governing laws for electromagnetism. In simulation, in order to guarantee that numerical solutions are physically feasible, it is essential to guarantee that the Gauss law of magnetism is still exactly satisfied. In other words, the magnetic field should be divergence-free even discretely to ensure that no magnetic monopoles exist. Such discretizations are called structure-preserving and are equipped with a discrete de Rham complex. In order to guarantee these properties even when using an iterative method, the development of robust multigrid preconditioners where the divergence-free magnetic field is preserved exactly at each time step is essential to guaranteeing physical solutions.

Publications

S. C. Brenner, C. Cavanaugh, L.-Y. Sung, A Hodge decomposition finite element method for an elliptic Maxwell boundary value problem on general polyhedral domains. Submitted.
S. C. Brenner, C. Cavanaugh, L.-Y. Sung, A Hodge decomposition finite element method for the quad-curl problem on polyhedral domains. J. Sci. Comput., 100(2024), pp. 1--35. Journal Link
J. H. Adler, C. Cavanaugh, X. Hu, A. Huang, and N. Trask, A stable mimetic finite-difference method for convection-dominated diffusion equations. SIAM J. Sci. Comput., 45 (2023), pp. A2973--A3000. Journal Link, Arxiv
J. H. Adler, C. Cavanaugh, X. Hu, and L. T. Zikatanov, A finite-element framework for a mimetic finite-difference discretization of Maxwell's equations, SIAM J. Sci. Comput., 43 (2021), pp. A2638--A2659. Journal Link, Arxiv

PhD Thesis: Structure-Preserving Discretizations for Partial Differential Equations. Link, pdf.

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