Scalable preconditioned conjugate gradient inversion of vector finite element mass matrices

Mass matrices arise in the numerical solution of time-dependent partial differential equations by the Galerkin method. Since these systems must be inverted at each time step, rapid inversion algorithms for these systems are important. When nodal finite elements are used as basis functions, it is known that the mass matrices can be consistently approximated by a diagonal matrix or solved by a scalable conjugate gradient method. This may not be the case for other basis functions. In this paper, we show that the preconditioned conjugate gradient method is scalable when used to invert mass matrices that arise from vector finite element basis functions. These basis functions are particularly important for solving Maxwell's equations on unstructured grids by the Galerkin method.