Abstract
We consider the numerical discretization of the time-domain Maxwell’s equations with an energy-conserving discontinuous Galerkin finite element formulation. This particular formulation allows for higher order approximations of the electric and magnetic field. Special emphasis is placed on an efficient implementation which is achieved by taking advantage of recurrence properties and the tensor-product structure of the chosen shape functions. These recurrences have been derived symbolically with computer algebra methods reminiscent of the holonomic systems approach.
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Notes
- 1.
These are polynomials which are products of univariate polynomials.
- 2.
Values at a boundary.
- 3.
Extensions to 3D are straightforward.
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Acknowledgements
We would like to thank Veronika Pillwein for making contact between the first- and the last-named author and for kindly supporting our work by interpreting between the languages of symbolics and numerics. Christoph Koutschan was supported by the Austrian Science Fund (FWF): SFB F013 and P20162-N18, and partially by NFS-DMS 0070567 as a postdoctoral fellow.
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Koutschan, C., Lehrenfeld, C., Schöberl, J. (2012). Computer Algebra Meets Finite Elements: An Efficient Implementation for Maxwell’s Equations. In: Langer, U., Paule, P. (eds) Numerical and Symbolic Scientific Computing. Texts & Monographs in Symbolic Computation. Springer, Vienna. https://doi.org/10.1007/978-3-7091-0794-2_6
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DOI: https://doi.org/10.1007/978-3-7091-0794-2_6
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