Publications

My academic publications and research papers.

• Coupled domain-boundary variational formulations for Hodge–Helmholtz operators

  E. Schulz, R. Hiptmair

  Integral Equations and Operator Theory, 2022

  We couple the mixed variational problem for the generalized Hodge-Helmholtz or Hodge-Laplace equation posed on a bounded three-dimensional Lipschitz domain with first-kind boundary integral equations arising when constant coefficients are assumed in the unbounded complement. Using recently developed Calderón projectors for symmetric coupling, we prove stability away from resonant frequencies by establishing a generalized Gårding inequality (T-coercivity). The resulting system describes the scattering of monochromatic electromagnetic waves at a bounded inhomogeneous isotropic body possibly having a rough surface.

  DOI arXiv

• First-kind boundary integral equations for the Dirac operator in 3-dimensional Lipschitz domains

  E. Schulz, R. Hiptmair

  SIAM Journal on Mathematical Analysis, 2022

  We develop novel first-kind boundary integral equations for Euclidean Dirac operators in 3D Lipschitz domains comprising square-integrable potentials and involving only weakly singular kernels. Generalized Gårding inequalities are derived and the boundary integral operators are shown to be Fredholm of index zero. Their finite-dimensional kernels are characterized and their dimension is equal to the sum of the Betti numbers of the domain's boundary.

  DOI arXiv

• Div–curl problems and H¹-regular stream functions in 3D Lipschitz domains

  M. Kirchhart, E. Schulz

  Mathematical Methods in the Applied Sciences, 2021

  We consider the problem of recovering the divergence-free velocity field of a given vorticity on a bounded Lipschitz domain. To that end, we solve the div-curl problem and express the solution in terms of a vector potential (stream function). After discussing existence and uniqueness of solutions and associated vector potentials, we propose a well-posed construction for the stream function. A numerical method based on this construction is presented, and experiments confirm that the resulting approximations display higher regularity than those of another common approach.

  DOI arXiv

• Convergence of discrete exterior calculus approximations for Poisson problems

  E. Schulz, G. Tsogtgerel

  Discrete & Computational Geometry, 2019

  Discrete exterior calculus (DEC) is a framework for constructing discrete versions of exterior differential calculus objects, widely used in computer graphics, computational topology, and discretizations of the Hodge-Laplace operator. However, a rigorous convergence analysis of DEC has always been lacking. We prove that DEC solutions to the scalar Poisson problem in arbitrary dimensions converge pointwise to the exact solution at least linearly with respect to the mesh size. Numerical experiments show that the convergence is in fact of second order when the solution is sufficiently regular.

  DOI arXiv