The numerical simulation of wave phenomena is the center of my research activity. I am concerned with the development and mathematical analysis of Galerkin methods and their implementation in the finite element library NGSolve, opens an external URL in a new window. My interests include different kinds of waves, e.g., in electromagnetics, acoustics and elastics, in frequency- as well as in time-domain. An up-to-date list of publications can be found under publications.

Fields of research are: unbounded domains and absorbing boundaries, fast time-domain solvers, Rresonance problems, and the implementation of the methods in NGSolve

In applications it is often necessary to study waves in domains which are unbounded or large compared to the respective wavelength. Many numerical methods including finite elements are based on the decomposition of the domain in question into many small bounded components. To apply these methods the problem has to be simulated on an artificially truncated domain. To avoid unwanted reflections on the artificial boundaries special boundary conditions have to be imposed. One possibility to do so is to surround the artificially truncated domain mathematically by a damping layer, without generating reflections. Such layers are called Perfectly Matched Layers (PMLs).

Publications adressing absorbing boundaries include

• M. Halla, M. Kachanovska, M. Wess, Radial perfectly matched layers and infinite elements for the anisotropic wave equation,  2024, [arXiv] [pdf]
• M. Halla, M. Kachaovska, M. Wess, Radial perfectly matched layers and infinite elements for the anisotropic wave equation, The 16th International Conference on Mathematical and Numerical Aspects of Wave Propagation (WAVES 2024), [doi], [ResearchGate], [pdf]
• É. Bécache, M. Kachanovska, M. Wess, Convergence analysis of time-domain PMLs for 2D electromagnetic wave propagation in dispersive waveguides, SAIM: Math. Modell. Numer. Anal., jul, 2023, [doi] [www] [pdf]
• É. Bécache, M. Kachanovska, M. Wess, Stability and convergence of time-domain perfectly matched layers in dispersive waveguides, 15th International Conference on Mathematical and Numerical Aspects of Wave Propagation, Palaiseau, France, July 2022, [ResearchGate]
• L. Nannen, M. Wess, Complex-scaled infinite elements for resonance problems in heterogeneous open systems, Advances in Computational Mathematics, vol. 48, dec, 2021, [doi] [www] [pdf]
• M. Wess, Frequency-dependent complex-scaled infinite elements for exterior Helmholtz resonance problems, Dissertation TU Wien, 2020, [reposiTUm] [pdf] [doi]
• B. Auinger, K. Hollaus, M. Leumüller, L. Nannen, M. Wess, Complex Scaled Infinite Elements for Electromagnetic Problems in Open Domains, 14th International Conference on Mathematical and Numerical Aspects of Wave Propagation, Vienna, Austria, Aug 2019, [doi] [reposiTUm]
• L. Nannen, K. Tichy, M. Wess, Complex Scaled Infinite Elements for Wave Equations in Heterogeneous Open Systems, 14th International Conference on Mathematical and Numerical Aspects of Wave Propagation, Vienna, Austria, Aug 2019, [doi] [pdf] [reposiTUm]

One possibilty to simulate time-domain waves is to discretize the underlying partial differential equation in space first. The resulting (very large) system of ordinary differential equations in time can be treated using established numerical methods. To this end, in every time-step a large system of linear equations has to be solved. For so-called explicit methods it is possible to obtain a system matrix (mass matrix) which has a very favourable form for inversion. Subject to my research are spacial discretizations which have to be adapted to the equations in question and yield favourable matrices for fast explicit methods.

Publications adressing time-domain solvers include

• L. Codecasa, B. Kapidani, J. Schöberl, M. Wess, Mass-lumped high-order cell methods for the time-dependent Maxwell's equations, The 16th International Conference on Mathematical and Numerical Aspects of Wave Propagation (WAVES 2024), [doi], [ResearchGate], [pdf]
• M. Wess, B. Kapidani, L. Codecasa, J. Schöberl, Mass lumping the dual cell method to arbitrary polynomial degree for acoustic and electromagnetic waves,  Journal of Computational Physics, vol. 513, 2024, [doi], [www], [pdf]

Resonance phenomena occur when a wave system is excited by certain (resonance-)frequecies and can be characterized by the fact that inserted energy is locally conserved. Thus an excitement by a resonace frequency may lead to the breakdown of the system, the so-called resonance catastrophe. Mathematically the computation of resonances leads to eigenvalue problems. Solving such problems is usually very demanding in terms of computational effort and used memory. Subject to my research are efficient algorithms for large eigenvalue problems which result from the discretization of resonance problems.

Publications adressing resonance problems include

• L. Nannen, M. Wess, A Krylov Eigenvalue Solver Based on Filtered Time Domain Solutions, 2024, [arXiv] [pdf]
• L. Nannen, M. Wess, Computing resonances of a wind instrument using a Krylov solver based on ltered
  time domain solutions, The 16th International Conference on Mathematical and Numerical Aspects of Wave Propagation (WAVES 2024), [doi], [ResearchGate], [pdf]
• M. Wess, Frequency-dependent complex-scaled infinite elements for exterior Helmholtz resonance problems, Dissertation TU Wien, 2020, [reposiTUm] [pdf] [doi]
• L. Nannen, M. Wess, Computing scattering resonances using perfectly matched layers with frequency dependent scaling functions, BIT. Numerical mathematics, vol. 58, pp. 373-395, 2018 [doi] [www] [pdf]

The open source library NGSolve, opens an external URL in a new window is developed in the CME group under the lead of Prof. Joachim Schöberl at the institute for Analysis und Scientific Computing and is an implementation of many variants of the finite element method for various physical applications. All numerical algorithms are implemented and tested in NGSolve or addon packages and are thus available for researchers and applicants.

Publications adressing implementation include

• M. Wess, J. Schöberl, The Dual Cell Method in NGSolve, Jupyter-Book online,  https://ngsolve.github.io/dcm