# 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.

# 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 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.

# Participation in the Electromagnetics — Modelling, Simulation, Control and Industrial Applications Workshop at WIAS, Berlin

The workshop takes place May 13 – 17, 2013 at WIAS, Berlin.

My talk will have the title
“Approaches to Time-Domain Electromagnetics with the Discontinuous Galerkin Method”
and will outline hp-adaptive meshes for DG time-domain and our new DG Trefftz method.

# Adaptive Simulation of a Folded Patch Antenna with Dynamical hp-meshes

This is another example of a dynamically adapted hp-mesh using SMOVE. The example is part of a paper I will submit soon (preprint link). It shows a triple slot patch antenna. The structure is included in the examples of CST MICROWAVE STUDIO.

To view the video, depending on your browser and platform either press Click To Play or just click the picture and allow for some seconds caching time. If the video doesn’t work (Quicktime plugin required + there appears to be a problem with Firefox) please use one of these direct video links [FLV lo-res (2.9 MB)] [FLV hi-res (9.2 MB)] [GIF lo-res (3.5 MB)] [GIF hi-res (9.2 MB)].

The absolute value of the electric field is on the left and the hp-mesh using embedded triangles for visualizing the tensor product orders (cf. Demkowicz, Solin) is on the right. The adaptation strategy is HP_ANISO, i.e., allowing for anisotropic refinement in the mesh step size and the polynomial order. The computation and adaptation is done in a three-dimensional domain. The viewplane is located at the bottom of the antenna substrate.

Electric farfield in the azimuth plane.

Electric farfield in the elevation plane.

# DFG granted project prolongation

The German Research Foundation DFG (Deutsche Forschungsgemeinschaft) positively evaluated the prolongation proposal for Martin Lilienthal‘s PhD project Autonome Steuerung von Gitteradaptionen im Rahmen der Diskontinuierlichen Galerkin Methode angewandt auf Probleme der Elektrodynamik (Autonomous Control of Mesh Adaptations for the Discontinuous Galerkin Method Applied to Problems in Electrodynamics).