Eigenvalue Solvers for Electromagnetic Fields in Cavities

We investigate algorithms for computing steady state electromagnetic waves in cavities. The Maxwell equations for the strength of the electric eld are solved by (1) a penalty method using common linear and quadratic node-based nite elements, and (2) a mixed method with linear and quadratic nite edge elements for the eld values and corresponding node-based nite elements for the Lagrange multiplier. These are two approaches that avoid so-called spurious modes which are introduced if the divergence free condition for the electric eld is not treated properly. The resulting large sparse matrix eigenvalue problems have been solved by various algorithms (1) subspace iteration, (2) block Lanczos algorithm, (3) implicitly restarted Lanczos algorithm and (4) Jacobi-Davidson algorithm. For all nite element approximations we compare the amount of work it takes each solver to compute a few of the smallest positive eigenvalues and corresponding eigenmodes to a given accuracy. 1. Introduction Most particle accelerators use standing waves in cavities to produce the high voltage RF elds required for the acceleration of the particles. The mathematical model for these high frequency electromagnetic elds is the eigenvalue problem solving the Maxwell equations in a bounded volume 28]. Usually, the eigenneld corresponding to the fundamental mode of the cavity is used as the accelerating eld. Due to higher harmonic components contained in the RF (radio frequency) power fed into the cavity, and, through interactions between the accelerated particles and the electromagnetic eld, higher order modes can be excited. The RF engineer designing such an accelerating cavity therefore needs a tool to compute the fundamental and about ten to twenty of the following eigenfrequencies together with the corresponding electromagnetic eigennelds. Historically, the attempts to solve such Maxwellian eigenvalue problems have very often suffered from so-called spurious modes that disturbed the searched eigenmodes, in particular with nite element approximations in three dimensions. A method invented between 1975 and 1980 by Weiland 59] 60], the so called nite integration technique (FIT), could completely avoid the problem of spurious modes, but at the price of going back to a nite diierence scheme. Real cavities mostly have smooth inner surfaces with exceptions at special points where the surface geometry can have ne structural details like e.g. at a coupling loop for feeding-in the RF power. Such strong variations of scale clearly favour nite element methods against nite diierence schemes. In the rst half of this paper we will review methods to …