Research Topic

Error control and hp-adaptivity for the time-dependent Maxwell equations

The aim of this project is to design an error-controlled, fully space-time hp-adaptive finite element method  for the time-dependent Maxwell system.


The accurate solution of large scale electromagnetic problems, where short wavelengths need to be resolved in large computational domains, remains challenging. Examples include antenna design, broadband scattering problems or electrically large structures. Especially for problems, where dispersion errors dominate, high order methods have advantages. Unfortunately high-order approximations alone are only effective, when the solution is smooth. Many problems, especially, when reentrant corners or jumps in the material coefficients are present, have in these regions non-smooth solutions. An attractive approach to deal with this kind of problems is to combine local high-order approximations in regions of the computational domain, where the solution is smooth with local mesh refinement in regions where the solution ins non-smooth. If this is done in a judicious way, it is possible to obtain exponential convergence, even for solutions, which are locally non-smooth. Within an adaptive method, this can lead to drastical savings in terms of degrees of freedom.

Results and Outlook

Space-time hp-Galerkin Method

Within the project I have designed a hp-Galerkin method [1], which allows for local hp-refinement in space as well as in time. 

Error control and adaptivity

A goal-oriented a posteriori error estimator has been developed, by extenting the Ideas of Becker and Rannacher [2] to the space-time discretization of Maxwell's equations. Based on the goal-oriented error estimator, hp-adaptive algorithms have been designed.


[1] Martin Lilienthal, Sascha M Schnepp, and Thomas Weiland, Non-dissipative space-time
$ hp $-discontinuous galerkin method for the time-dependent maxwell equations, arXiv
preprint arXiv:1307.5310, (2013).

[2] R. Becker and R. Rannacher, An optimal control approach to a posteriori error estimation
in finite element methods, Acta numerica, 10 (2001), p. 1102.



hp-adaptive simulation of the scattering from a dielectric sphere.From left to right: distribution of |Eh|, spatial polynomial degrees (px, py, pz) and temporal refinement level.

Key Research Area

Computational electromagnetics


Martin Lilienthal


Dolivostra├če 15

D-64293 Darmstadt



+49 6151 16 - 24401 or 24402


+49 6151 16 - 24404




lilienthal (at) gsc.tu...

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