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 visible) after emission from a (simplified) metallic nanotip by an impinging femtosecond laser pulse impinging.

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.