Method for solution of Maxwell's equations in non-uniform media

What is it about?

It is shown that the L 2 norm of electric currents induced in a dissipative medium can never exceed the norm of the external currents. This allows the construction of a simple iteration method to solve Maxwell's equations. The method produces a series converging to the solution for an arbitrary conductivity distribution and arbitrary frequency of field variations. The convergence is slow if the lateral contrast of the conductivity distribution is about 104 or higher. A modification significantly improving the convergence is described in this paper. As an example, electromagnetic fields induced in the model (including the western part of" the Northern American continent and the adjacent part of the Pacific Ocean) are calculated.

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Why is it important?

The approach reduces Maxwell's equation to an integral equation with a contraction kernel. The equation may be solved by a simple iteration. Alternatively, it may be used a perfect preconditioner after applying which a conjugate gradient method may be used for even a faster solution. The approach may be applied to any heterogeneous medium which dissipates the energy of the electromagnetic field. It may be applied not only to the integral equation.