The largest eigen-problem solved by Omega3P has 93 million Degrees of Freedom (DOF). The problem was raised in computing wakefields of a 55-cell tapered structure, named H60VG3, which is considered the base line design for the Next Linear Collider.

T3P will utilize the finite element software framework already in place for the parallel finite element eigensolver Omega3P, and will follow the workflow that presently exists in Tau3P. Two solution approaches are being considered. In either case, we solve a single time-dependent Maxwell wave equation instead of advancing the two Maxwell curl equations separately as is done in Tau3P. One approach uses an implicit scheme based on the Newmark method. With proper choice of coefficients, the Newmark algorithm is shown to be unconditionally stable and therefore permits timesteps larger than those allowed by the Courant condition. However, matrix inversion is needed at every timestep as the mass matrix is no longer diagonal. As a result, the performance of TP is critically dependent on a good preconditioner for the linear system.

The other approach is to introduce orthogonal vector basis functions so that the mass matrix becomes diagonal and an explicit scheme can be used. Although only linear basis functions have been derived thus far, our goal is to extend the method to include quadratic basis functions for improved accuracy. We will compare the performance and accuracy of this explicit higher order scheme with those obtained with the implicit scheme based on quadratic finite elements.

Work is in progress to develop the implicit module of T3P and preliminary runs are expected soon on a simple problem in which a cavity is excited by a dipole antenna. This model will serve as a testbed for subsequent implementations of various preconditioning techniques as well as the development of the explicit module that involves orthogonal vector basis functions.