## Preconditionning harmonic Maxwell integral equation.

#### Jean-Claude NedelecEcole Polytechnique, Palaiseau France/CMAP

waves by objects. It concerns in particular harmonic acoustic problems with a growing demand from the car industry or the aeronautic industry in order to reduce the noise produce by engines (inside The integral technique are nowadays largely used to solve the linear problem of di_raction of or outside). It concerns also harmonic electromagnetic problems coming from the telecommunication industry, which need to design more and more complex antennas or other devices. One of the key number in these numerical techniques, is the size of the object compare to the wave length. Every numerical technique need to resolve the wavelength. This implies a certain number of degrees of liberty per wavelength. For instance, in order to solve the EFIE integral technique for Maxwell, using the classical edge element (RWG basis), it is observed (no proof) that 5 points per wavelength is necessary (and su_sant in most situations). It can go to 10 points per wavelength with a less precise technique such as collocation. The integral equation is written on the boundary of the object which is a 2D surface in the case of a 3D problem. Let d be its size and k the wave number. The number of unknowns to cover the surface increases as (dk)2. Then if we use adirect solver, the count of operations increases as (dk)6. We also need to store the full complex matrix associated which size increases as (dk)4. One of the technique
to overcome this problem consist of not building the matrix A, but instead to compute for every given unknown J the action AJ via an approximation which uses less memory and is more rapid.
Some researchers claim that wavelet techniques is the best tool to achieve this goal.
But the most largely used and successfull technique is the Fast Multipole Method. Then, if N is the number of unknowns, using a multilevel FMM technique, thecomputation of AJ requires only NLog(N) operations and the error introduced by this approximation can be well controlled.
But, now we cannot use a direct solver as we have not at our disposal the matrix A. So iterative
techniques are necessary. The most popular ones belong to the family of conjuguate gradient technique such as GMRES. Now the count of operation is NitNLog(N) if Nit is the number of iterations. Here appears the necessity of preconditionning as Nit can be large or in_nite.
We have observed that the number of iterations is heavily dependent on the preconditionner but also on several others parameters such as { the wave number and the choice of the integral formulations { the nature and the shape of the object (PEC, dielectric, cavity, plate,_ _ _)
Engquist 60th 27/08/2005
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