Publications

My academic publications and research papers.

• Boundary integral exterior calculus

  E. Schulz, R. Hiptmair, S. Kurz

  Journal of the European Mathematical Society, 2025

  We report a surprising and deep structural property of boundary integral operators occurring in first-kind boundary integral equations associated with Hodge–Dirac and Hodge–Laplace operators for de Rham Hilbert complexes on a bounded domain Ω in a Riemannian manifold. We show that, as regards their induced bilinear forms, those boundary integral operators are Hodge–Dirac and Hodge–Laplace operators in the weak sense, this time set in a trace de Rham Hilbert complex on the boundary ∂Ω whose underlying spaces of differential forms are equipped with non-local inner products defined through layer potentials. On the way to this main result we conduct a thorough analysis of layer potentials in operator-induced trace spaces and derive representation formulas.

  DOI PDF

• Minimal ℓ² norm discrete multiplier method

  E. Schulz, A. T. S. Wan

  Journal of Computational Dynamics, 2024

  We introduce the Minimal ℓ² Norm Discrete Multiplier Method (MN-DMM), an extension to the Discrete Multiplier Method where conservative finite difference schemes for dynamical systems with multiple conserved quantities are constructed procedurally. MN-DMM utilizes the right Moore-Penrose pseudoinverse of the discrete multiplier matrix to solve an underdetermined least-squares problem associated with the discrete multiplier conditions. We prove consistency and conservative properties of MN-DMM schemes and demonstrate their applicability on various problems including the planar restricted three-body problem, Lorenz system, and geodesic curves on Schwarzschild spacetime.

  DOI arXiv

• Traces for Hilbert complexes

  R. Hiptmair, D. Pauly, E. Schulz

  Journal of Functional Analysis, 2023

  We study a new notion of trace operators and trace spaces for abstract Hilbert complexes. We introduce trace spaces as quotient spaces and annihilators, characterize the kernels and images of the related trace operators, and discuss duality relationships between trace spaces. We show that many properties of classical boundary traces associated with the Euclidean de Rham complex on bounded Lipschitz domains are rooted in the general structure of Hilbert complexes. We establish that if a Hilbert complex admits stable regular decompositions with compact lifting operators, the associated trace Hilbert complex is Fredholm.

  DOI arXiv

• Spurious resonances in coupled domain-boundary variational formulations of transmission problems in electromagnetism and acoustics

  E. Schulz, R. Hiptmair

  Computational Methods in Applied Mathematics, 2022

  We develop a framework shedding light on common features of coupled variational formulations arising in electromagnetic scattering and acoustics. We show that spurious resonances haunting coupled domain-boundary formulations based on direct boundary integral equations of the first kind originate from the formal structure of their Calderón identities. We explicitly characterize the solution space of the coupled problem and prove that it completely vanishes under the exterior representation formula.

  DOI