This paper presents the advances in electromagnetic (EM) field computation that have been enabled by the US DOE SciDAC Accelerator Science and Technology project which supports the development and application of a suite of electromagnetic codes based on the higher-order finite element method. Implemented on distributed memory supercomputers, this state of the art simulation capability has produced results which are of great interest to accelerator designers and with realism previously not possible with standard codes. Examples from work on the International Linear Collider (ILC) project are described.