Binary black hole simulation with an adaptive finite element method: Solving the Einstein constraint equations

Abstract

In this paper we start a systematic investigation of applying an adaptive finite element method to the Einstein equations, especially binary compact object simulations. To our knowledge, this is the first study on this topic. Puncture type initial data are solved with the adaptive finite element method. The numerical scheme proposed in the current work can be straightforwardly extended to the general type of initial data of the Einstein equations. The Parallel Hierarchical Grid library and the existing numerical relativity code AMSS-NCKU are used to develop the adaptive finite element Einstein solver. In the unsmooth toy model problem, the adaptive mesh refinement operation can catch the unsmooth region efficiently. The numerical solution deviates the exact solution by an error less than $1{0}^{\ensuremath{-}5}$. In the binary black hole problem, our solution is consistent with the one gotten by the TwoPuncture code which uses a pseudospectral method. As we expected, the solution gotten by the finite element method is less accurate than that gotten by the spectral method. But the relative error distributes almost uniformly. The adaptive mesh refinement method is quite efficient and it does not waste computational effort. Our finite element code is more flexible than the TwoPuncture code. It can be used to treat other general initial data problems such as the three black holes problem, besides the binary black hole problem. We test one typical three black holes problem also. In all of the test cases, our adaptive finite element code works quite well.