Niklas Wellander

Department of Mathematics

University of California

Santa Barbara, CA 93106

USA

If the typical spatial length scale in a heterogeneous
material is much smaller than the wavelength of an electromagnetic
field then the field cannot resolve the fine scales. In numerical
implementations the fine scales require a numerical mesh which is far
too large for any computer. One way to take care of that problem is to
homogenize the Maxwell equations, i.e. to find the effective material
properties of the heterogeneous material. The effective properties
correspond to a homogeneous material which is a good approximation of
the heterogeneous material in the sense that the solutions of the
homogenized equations are good approximations of the solutions of the
original equations. The effective material properties are obtained by
solving local problems on the unit cell and taking suitable averages.
I will give examples of how the Maxwell equations can be homogenized,
using two-scale convergence, in some linear and nonlinear cases. In
particular I will address homogenization of varistor ceramics,
nonlinear resistors used as devices for protection against surges in
power lines.