Abstract
We investigate algorithms for computing steady state electromagnetic waves in cavities. The Maxwell equations for the strength of the electric field are solved by a mixed method with quadratic finite edge (N\'ed\'elec) elements for the field values and corresponding node-based finite elements for the Lagrange multiplier. This approach avoids so-called spurious modes which are introduced if the divergence-free condition for the electric field is not treated properly. To compute a few of the smallest positive eigenvalues and corresponding eigenmodes of the resulting large sparse matrix eigenvalue problems, two algorithms have been used: the implicitly restarted Lanczos algorithm and the Jacobi-Davidson algorithm, both with shift-and-invert spectral transformation. Two-level hierarchical basis preconditioners have been employed for the iterative solution of the resulting systems of equations.
Highlights
The common way to produce the accelerating electromagnetic fields in cyclic accelerators is to excite standing waves in accelerating cavities
By using numerical experiments reported in [9], we identified Sorensen’s implicitly restarted Lanczos (IRL) algorithm [10] and the Jacobi-Davidson (JD) algorithm [11] to be the algorithms best suited to solve the large sparse eigenvalue problems that arise with the above finite element discretizations
The Jacobi-Davidson algorithm is the faster eigensolver than the implicitly restarted Lanczos algorithm. This is due to the fact that IRL executes many more steps in the inner iteration as the Lanczos vectors have to be computed to high accuracy
Summary
The common way to produce the accelerating electromagnetic fields in cyclic accelerators is to excite standing waves in accelerating cavities. By using numerical experiments reported in [9], we identified Sorensen’s implicitly restarted Lanczos (IRL) algorithm [10] and the Jacobi-Davidson (JD) algorithm [11] to be the algorithms best suited to solve the large sparse eigenvalue problems that arise with the above finite element discretizations. They are to be preferred to subspace iteration [12] or the block Lanczos algorithm [13]. VII we compare the performance of several variants in solving the model problem
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