Adaptive error estimators for electromagnetic field solvers

Under the U.S. Department of Energy's Scientific Discovery through Advanced Computing (SciDAC) program, the Advanced Computations Department (ACD) at Stanford Linear Accelerator Center (SLAC) has been developing advanced numerical tools for high performance and high accuracy linear accelerator simulations. One of the most important aspects in accelerator simulations is the accurate calculation of some low frequency electromagnetic fields within accelerator cavities, i.e., accurate eigensolutions of the frequency domain Maxwell's equations.;Adaptive mesh refinement is a method of dynamic meshing. The advantages of a dynamic meshing scheme are: saving computational resources over a static mesh approach, saving storage over a static mesh approach and having complete control of mesh resolution, compared to the fixed resolution of a static mesh approach. Essential to adaptive mesh refinement are error estimators which compute the error estimates based on the computed numerical solutions. The so-called h refinement corresponds to the refinement of mesh spacing similar to the finite difference and finite volume methods. The so-called p refinement corresponds to the enrichment of the polynomial orders of the elements.;The objective of this work is to develop an h error estimator and a p error estimator for the adaptive mesh refinement for electromagnetic field solvers using three-dimensional vector finite element methods. A domain decomposition strategy is employed to compute the error estimates in parallel. The computational domain of the accelerator structure is partitioned equally in terms of number of bases functions.;A three-dimensional vector finite element solver has been developed in ACD to simulate electromagnetic fields in linear accelerator structures. In the finite element method, both spacing of the mesh and orders of elements determine the accuracy and the cost of the computation. Fine mesh spacing is usually needed in the difficult regions due to steep gradients, shocks, discontinuities, etc. Uniformly fine meshes are computationally costly and in some cases the optimum spacing cannot be obtained a priori.