Simulating Black Hole Formation in Vacuum with the Finite Element Method

Critical phenomena in gravitational collapse involves the study of what occurs at the threshold between black hole formation and field dispersal. Most notably, this phenomena provides an interesting test of cosmic censorship, and vacuum collapse (implosion of gravitational waves) provides a further test that doesn't depend on an imposed field but rather is inherent to general relativity itself. Computationally, these simulations provide many challenges that the Finite Element Method (FEM) may be well suited to address. Namely, FEM more evenly distributes error across the mesh, can adaptively refine the mesh to better fit the data similar, and in some ways superior, to finite difference methods, and allows for higher order basis functions similar to spectral methods which often lead to lower overall degrees of freedom and less mesh refinements leading to faster overall evolutions. Some results are presented from efforts to make a stable FEM vacuum collapse simulation.