Research

Commuting projections are a fundamental tool in FEEC. They map scalar and vector-valued functions onto finite element spaces in a way that commutes with the differential operators. They are the standard route to proving stability and error estimates for mixed and structure-preserving finite element methods. The most useful projections are local, bounded in L², and commuting all at once. Having these three properties in combination is hard to arrange. The “canonical” interpolants commute but need extra smoothness, while earlier bounded projections were either global or bounded only in a stronger norm.

With Alexandre Ern, Johnny Guzmán, and Martin Vohralík, we constructed projections with all three properties onto the Lagrange, Nédélec, Raviart–Thomas, and discontinuous spaces of lowest and higher order on tetrahedral meshes. The projections are assembled from local weight functions which are piecewise polynomials on the Alfeld split of a patch of elements, computed by solving small discrete problems whose stability comes from discrete Poincaré inequalities (for the divergence and curl as well as the gradient).

The needed discrete Poincaré inequalities are often cited as known, but the results are spread across the literature under different hypotheses. Of interest for us, was having these inequalities with uniformly bounded constants over patches of elements. With Alexandre Ern, Johnny Guzmán, and Martin Vohralík, we wrote a review on these inequalities, providing three routes to proving the inequalities with ideas for tracking how the constants depend on the shape regularity of the mesh, the number of elements, and the polynomial degree, with particular attention to local patches of elements.

Furthermore, with Alexandre Ern and Johnny Guzmán, we developed commuting projections that also preserve piecewise polynomial boundary traces, which are useful for problems with nonhomogeneous boundary data.

DEC represents discrete differential forms as cochains. These are essentially numbers attached to the vertices, edges, faces, and cells of a simplicial mesh and of its dual mesh. Then one has a discrete exterior derivative as the incidence matrix (coboundary operator) of the simplicial mesh and discrete Hodge stars are built from the geometry of the two meshes. DEC has wide applications in computer-aided geometric design as well as numerical methods for PDEs. The convergence theory for DEC on the other hand has lagged behind its application.

With Johnny Guzmán, we developed an analysis framework for DEC convergence and demonstrated its use by analyzing convergence for the Hodge-Laplace PDE. We also made connections between DEC and FEEC by viewing primal cochains as Whitney forms, and dual cochains as generalized Whitney forms on the circumcentric dual mesh.

The Poincaré lemma is a well known result in differential geometry which states that closed differential forms are exact on contractible domains. The standard proof is constructive through the use of a so-called Poincaré operator. In DEC and FEEC, discrete differential forms are realized as cochains and Whitney forms respectively. Thus, a natural question is to develop discrete Poincaré operators which map closed discrete differential forms on a contractible domain to exact discrete differential forms. With Johnny Guzmán, Anil Hirani, and Bingyan Liu, we constructed such operators which provide numerical methods for computing grad, curl, and div potentials. We also developed discrete Bogovskiĭ operators, which are analogues to the Poincaré operators preserving homogeneous boundary conditions. These operators also have applications in understanding the discrete wedge product of DEC. In further work with Johnny Guzmán and K. Olav Tyssvang, we extended these ideas to higher polynomial order Whitney forms.

I am also interested in the fundamental theory behind splines on triangulations. Such splines are used in finite element methods and in computer-aided geometric design. A fundamental question posed by Strang in the 1970s asked: What is the dimension of the space of C¹ piecewise polynomials of degree d on a given planar triangulation? The answer to Strang's question turns out to depend on both the combinatorial structure of the mesh as well as its geometry. Strang conjectured a dimension formula for these spline spaces and this formula was shown to hold for polynomial degree four and higher, and to not hold for degree two. The cubic case was a long-standing open problem which was resolved recently by Wu and Deng (2026) who exhibited a triangulation which breaks Strang's formula.

In a recent preprint, I showed how recent Large Language Models (LLMs) could be used to set up systematic autonomous searches for such triangulations which break Strang's formula. The key turned out to involve the combinatorics of symmetric triangulations, and I developed an algorithm which was able to find new triangulations breaking Strang's formula.