The MULTIMODE code was initially developed for computing the lowest eigenfrequences and electromagnetic fields in homogeneous waveguides and axyally symmetric cavities:
A. Fedoseyev, V. Gusev, M. Kaschiev, V. Kaschieva, V. Paramonov, MULTIMODE - a powerfull code for frequency spectrum computation of electromagnetic fields in axially symmetric cavities and longitudinally homogeneous waveguides of arbitrary shape. Nuclear Instrument and Methods in Physics Research, 227 (1984) 411-419.

Further developments extended the MULTIMODE code to the analysis of electromagnetic fields with asimuthal variations and fields in periodic accelerating structures:
(i) V. Anikeyeva, I. Gonin, M. Kaschiev, V.Paramonov, A. Fedoseyev. Mathematical Modelling of Electromagnetic Fields with Azimuthal Variations in Axially Symmetric Cavities. Formulation of the Problem and Solution Methods. Preprint IHEP 84-92, Serpukhov, 1984, 16p.
(ii)V. Anikeeva, A. Fedoseyev. Solution Algorithm for Generalized Nonsymmetric Eigenvalue problems. Preprint IHEP 85-24, Serpukhov, 1985, 11p.
(iii)A. Fedoseyev. Subspace Iteration Method for Unsymmetric Eigenvalue Problems. Preprint IHEP 85-25, Serpukhov, 1985, 11p.
(iii) I. V. Gonin, L. V. Kravchuk, V.V. Paramonov, A.I. Fedoseyev et al. Some Methods of the Stop-Band Width Evaluation in the Compensated Accelerating Structures, IEEE Transactions on Nuclear Science, 5, NS-32 (1985) 2918-2921.

A problem formulation (derived from Maxwell equations) for axially symmetric solution case and longitudinally homogeneous waveguides of arbitrary shape is here. A finite element method with 8-noded isoparametric elements is used. Superelement mapping technique is used for complicated domain geometries:

· To find simultaneously several electromagnetic field modes and frequencies we used the subspace iteration or Davidson methods for both generalized symmetric and non-symmetric eigenproblems. Typical accuracy (confirmed in the experiments) is 0.001% to 0.000001% for RF-frequencies (the better if the finer mesh is used).
Here is a picture of electromagnetic field for the third mode in the DAW structure:

and here one is for the 6-th mode:
.

The next example is a mesh for the two-chamber H-waveguide

The central part of the region is shown in details:

· A 2D and 3D picture of electric field for the first three modes for the H-waveguide (click on picture for a better resolution image):

· The MULTIMODE code includes few different automatic 2D mesh generators and postprocessors.

TYPICAL MULTIMODE RUN CHARACTERISTICS
EXAMPLE 1. Cylindrical DAW structure shown above: a computer time (Pentium-133 PC) for computation of seven modes with an accuracy in frequency of 0.0002 % is about 35 secs.

EXAMPLE 2. H-waveguide shown above: computer time (Pentium-133 PC) for computation of all three modes is 20 secs.