__Convergence of an ADI splitting for Maxwell's equations__

M. Hochbruck, T. Jahnke and R. Schnaubelt

**Abstract**

Maxwell's equations provide the foundation for the theory of
electromagnetism, and solving these equations numerically is an
important task in many applications. For problems posed on a cuboid
or on $\mathbb{R}^3$, the alternating direction implicit method
proposed by Zheng, Chen, and Zhang (2000) is particularly attractive,
because this method is unconditionally stable and computationally
cheap. The main idea is, roughly speaking, to decompose the Maxwell
operator in such a way that the sub-flows can be propagated in a
stable and efficient way.
In this talk, second-order convergence for the semi-discretization in
time is shown in the framework of operator semigroup theory. The
proof is based on results concerning the regularity of the Cauchy
problems of the sub-flows, which then allow to apply an abstract
convergence proof by Hansen and Ostermann [1].
Before the error analysis, well-posedness of Maxwell's equations on
cuboids and on $\mathbb{R}^3$ will be discussed.

**Bibliography**

[1]
E. Hansen and A. Ostermann, Dimension splitting for evolution
equations, Numerische Mathematik, 108 (2008), pp. 557-570.