Research

A significant focus of my current research lies in structure-preserving and compatible discretization, with a particular emphasis on Finite Element Exterior Calculus (FEEC). This field, however, extends beyond its name: “finite element” may extend to discrete patterns like finite/lattice differences, currents, or networks, while “exterior calculus” can integrate Cartan’s full formalism (particularly with connections to differential geometry) or, more broadly, tensor analysis. The Bernstein-Gelfand-Gelfand (BGG) construction and representation theory are languages for expressing tensor symmetries.

Some specific topics I am interested in now include:

  • General theories of FEEC and Finite Element Tensor Calculus: This includes tensor finite elements (discretizing geometric objects such as metrics, curvature, torsion, connection forms, form-valued forms, double forms, or bundle-valued forms), links to discrete differential geometry, BGG machinery and representation theory. Construction of finite elements and complexes (connections to spline theories), high-order methods and solvers in the finite element context.

  • Computational topological hydrodynamics and magnetohydrodynamics (MHD): Topology-preserving numerical methods for fluid and plasma systems, relaxation problems (self-organization), dynamo problems and applications in solar physics, astrophysics, and fusion energy.

  • Algebraic and geometric modeling of generalized continua, discretization and solvers: This involves micropolar continua, defect theories, and multidimensional models, drawing inspiration from rational mechanics (e.g., the Cosserat brothers’ models, Eringen’s micropolar theory, Kröner and Nye’s defect modeling, and Yavari-Goriely’s geometric dislocation/disclination theory). Hilbert complexes and BGG constructions offer computationally friendly tools with built-in analysis.

  • Numerical relativity: Solving linearized and nonlinear Einstein equations numerically, with applications in astrophysics and gravitational wave sciences.

  • Discrete differential geometry: Christiansen’s pioneering work of interpreting Regge calculus as a finite element opens a door for designing, analysing, and generalising discrete geometric patterns from the finite element point of view. In addition, I am interested in a PDE analysis perspective, and interactions and applications with computer graphics, discrete physics (Lattice Gauge Theory, quantum and numerical gravity), and materials (origami etc.)

  • Geometry, topology, PDEs, and physics on graphs: The broad idea is to discretize continuous theories on graphs or networks. Finite Element Exterior Calculus (in the narrow sense) is a special case on triangulation. In fact, for differential forms, Finite Element Exterior Calculus and graph (co)homology and Hodge-Laplacian share many common definitions and properties. Other problems include random graphs and phase transition (statistical topology) and topological/geometric data analysis etc.

Although my perspective on these topics continually evolves, I maintain a consistent interest in investigating continuous and discrete structures, using them to design reliable, efficient, and high-fidelity numerical methods, and investigating the interactions between numerical analysis (finite elements) and other areas.

These slides summarized part of my current research interests.