# Paper on a Non-dissipative space–time hp-discontinuous Galerkin method for the time-dependent Maxwell equations published in JCP

Our paper dealing with the PhD project of Martin Lilienthal has been published in JCP: Non-dissipative space–time hp-discontinuous Galerkin method for the time-dependent Maxwell equations. The abstract reads:

A finite element method for the solution of the time-dependent Maxwell equations in mixed form is presented. The method allows for local hp-refinement in space and in time. To this end, a space–time Galerkin approach is employed. In contrast to the space–time DG method introduced in [Van der Vegt, JCP(182) 2002] test and trial spaces do not coincide. This allows for obtaining a non-dissipative method. To obtain an efficient implementation, a hierarchical tensor product basis in space and time is proposed. This allows to evaluate the local residual with a complexity of $$\mathcal{O}(p^4)$$ and $$\mathcal{O}(p^5)$$ for affine and non-affine elements, respectively.

# New Job from August 2014

I am starting a new position tomorrow — and will convert into a geophysicist (watch this for some background info). These are my new utensils:

Ok, seriously:
I wish to thank the Alexander von Humboldt-Foundation for supporting me over the last two years.
Particularly, I wish to thank Prof. Christian Hafner and the Institute of Electromagnetic Fields at ETH Zurich. Prof. Hafner was as welcoming as anyone can possibly be. Experiencing how he still enjoys his research after more than 35 years made a strong impression on me. Thank you!

Starting from August I am with the Geophysical Fluid Dynamics group of Prof. Tackley at ETH. They give me the opportunity to join the GeoPC project, which deals with infrastructure development for hybrid parallel smoothers for multigrid preconditioners. GeoPC is part of the larger PASC (Platform for Advanced Scientific Computing) initiative. The abstract of the project proposal reads:

The GeoPC project is developing computational infrastructure to enable massively parallel, scalable smoothers and coarse grid solvers to be used within multigrid preconditioners. This infrastructure is intended to (i) facilitate the execution of high resolution, 3D geodynamic models of the planetary evolution; (ii) provide the Earth Science community (and others) with a suite of continually maintained and re-usable HPC components to build robust multi-level preconditioners (iii) position Swiss computational geosciences in the emerging exa-scale era.

GeoPC is a PASC co-design project involving the University of Lugano (USI), the Swiss Federal Institute of Technology (ETH), University of Chicago (UC), Vienna University of Technology (TU Wien), as well as other stakeholders.

Our developments will become part of PETSc, the Portable, Extensible Toolkit for
Scientific Computation. I am very much looking forward to contributing to a library as renowned as PETSc!

My new coordinates are:
ETH Zurich
Institute of Geophysics – Geophysical Fluid Dynamics
NO H23, Sonneggstrasse 5, 8092 Zurich, Switzerland
Phone: +41 44 632 0244

# Student Paper Competition Award for Fritz Kretzschmar

Fritz Kretzschmar received a Student Paper Competition Award for his contribution entitled

The Discontinuous Galerkin Trefftz Method

at the 12th International Workshop on Finite Elements for Microwave Engineering (FEM2014). Coauthors are Igor Tsukerman, Thomas Weiland and myself. Congratulations!

If you are interested in more information on the time-domain DG Trefftz method have a look at our recent paper in JCAM and/or contact Fritz or me.

# Introduction to Computational Electromagnetics

The slides of today’s lecture on the Discontinuous Galerkin Method are available here (14 MB).

If you are interested in a project work or a bachelor/master thesis send me an email or come by.

# Paper on Discontinuous Galerkin methods with Trefftz approximations published in JCAM

Fritz Kretzschmar, Igor Tsukerman and I have a new paper in Journal of Computational Applied Mathematics (JCAM). The paper abstract reads

We present a novel Discontinuous Galerkin Finite Element Method for wave propagation problems. The method employs space–time Trefftz-type basis functions that satisfy the underlying partial differential equations and the respective interface boundary conditions exactly in an element-wise fashion. The basis functions can be of arbitrary high order, and we demonstrate spectral convergence in the $$L_2$$-norm. In this context, spectral convergence is obtained with respect to the approximation error in the entire space–time domain of interest, i.e. in space and time simultaneously. Formulating the approximation in terms of a space–time Trefftz basis makes high order time integration an inherent property of the method and clearly sets it apart from methods, that employ a high order approximation in space only.

# Paper on Error-Driven Dynamical hp-Meshes for the Discontinuous Galerkin Method in Time-Domain

My paper on error-driven dynamical hp-meshes for DG in the time-domain was accepted for publication in the Journal of Computational and Applied Mathematics. The paper abstract reads:

An hp-adaptive Discontinuous Galerkin Method for electromagnetic wave propagation phenomena in the time-domain is proposed. The method is highly efficient and allows for the first time the adaptive full-wave simulation of large, time-dependent problems in three-dimensional space. Refinement is performed anisotropically in the approximation order p and the mesh step size h regardless of the resulting level of hanging nodes. For guiding the adaptation process a variant of the concept of reference solutions with largely reduced computational costs is proposed. The computational mesh is adapted such that a given error tolerance is respected throughout the entire time-domain simulation.

For examples of dynamical hp-meshes see this post or this one. Please find the paper here on the JCAM website.

# Paper on the Numerical Modeling of Metallic Nano-Tips

Josip Mihaljevic, Jens Niegemann, Christian Hafner and I submitted a paper on the numerical modeling of metallic nano-tips to Quantum Matter. My contribution deals with the simulation of laser-induced electron emission from such tips. A visualization of the process is shown below.

Snapshots of electrons in an electrostatic field (not visualized) after emission from a (simplified) metallic nano-tip induced by an impinging femtosecond laser pulse.

UPDATE (Nov 3, 2013):
The paper has been accepted for publication.

UPDATE
The paper is available online.

# Paper on a Non-dissipative space-time hp-discontinuous Galerkin method for the time-dependent Maxwell Equations

Martin Lilienthal developed a space-time discontinuous Galerkin method. Unlike other works, this formulation is non-dissipative, which is achieved through non-coinciding choices of the trial and test spaces. Please find the preprint on arXiv.