By Brehm U., Kuhnel W.

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**Example text**

18) is only second-order accurate inside the PML. Never the less, based on [47], we expect the global error to be higher than second order. 6 Discrete regularization Model problem 1 is mainly of mathematical importance. Hence, we introduce model problem 2, which can be considered as a model for the electromagnetic wave scattering by a physical body. 17). A natural way to approximate the piecewise continuous function ε(x) is by using a high order polynomial interpolation. We do not require higher than second order of differentiability for the regularized function, because the analytic solution has only one continuous derivative at the interface.

Homogeneous) and in bodies which may be inhomogeneous media. For instance, a cellular phone sends signals from a building to the closest antenna to register its location. Another example is a sensor that emits electromagnetic pulses into the ground to check for land mines. These can be simulated by the solution of Maxwell equations with discontinuous coefficients. A discontinuity in coefficients represents propagation of electromagnetic waves between media with different dielectric and magnetic properties.

3) inside the PML include non-differentiated terms. In [26] it is shown that this can lead to the increase of the magnitude of the solution and finally to the overflow after a number of iterations. 32 In the leapfrog scheme this problem can be avoided by the exponential timedifferencing (see, for example [58]). 7): 1 ∂Px σy + Px = δPx ∂t ε ε ∂Px ∂Ex + σx Px = + σz Ex ∂t ∂t where δPx = ∂Hz ∂y − ∂Hy . ∂z Both equations can be rewritten in the following way: σy t ∂ e ε Px ∂t ∂ eσx t Px e−σx t ∂t e− σy t ε 1 = δPx ε = e−σz t ∂ eσz t Ex ∂t Without non-differentiable terms these equations can be integrated numerically.

### 15-vertex triangulations of an 8-manifold by Brehm U., Kuhnel W.

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