Solving the Einstein constraint equations on multi-block triangulations using finite element methods

Abstract

In order to generate initial data for nonlinear relativistic simulations, one needs to solve the Einstein constraints, which can be cast into a coupled set of nonlinear elliptic equations. Here we present an approach for solving these equations on three-dimensional multi-block domains using finite element methods. We illustrate our approach on a simple example of Brill wave initial data, with the constraints reducing to a single linear elliptic equation for the conformal factor $\psi$. We use quadratic Lagrange elements on semi-structured simplicial meshes, obtained by triangulation of multi-block grids. In the case of uniform refinement the scheme is superconvergent at most mesh vertices, due to local symmetry of the finite element basis with respect to local spatial inversions. We show that in the superconvergent case subsequent unstructured mesh refinements do not improve the quality of our initial data. As proof of concept that this approach is feasible for generating multi-block initial data in three dimensions, after constructing the initial data we evolve them in time using a high order finite-differencing multi-block approach and extract the gravitational waves from the numerical solution.

Changes made to match the version to appear in Classical and Quantum Gravity

References in corpus (13)

• High-accuracy waveforms for binary black hole inspiral, merger, and ringdown
• Spectral Methods for Numerical Relativity
• Solving Einstein's Equations With Dual Coordinate Frames
• A multi-block infrastructure for three-dimensional time-dependent numerical relativity
• A numerical study of the quasinormal mode excitation of Kerr black holes
• Rough solutions of the Einstein constraints on closed manifolds without near-CMC conditions
• Far-from-constant mean curvature solutions of Einstein's constraint equations with positive Yamabe metrics
• Multi-patch methods in general relativistic astrophysics - I. Hydrodynamical flows on fixed backgrounds
• How far away is far enough for extracting numerical waveforms, and how much do they depend on the extraction method?
• A new spectral apparent horizon finder for 3D numerical relativity
• Rough solutions of the Einstein constraint equations with nonconstant mean curvature
• Generating Initial Data in General Relativity using Adaptive Finite Element Methods
• Evolving relativistic fluid spacetimes using pseudospectral methods and finite differencing

Cited by in corpus (6)

• Stability of general-relativistic accretion disks
• Characteristic Evolution and Matching
• From Geometry to Numerics: interdisciplinary aspects in mathematical and numerical relativity
• An Operator-Based Local Discontinuous Galerkin Method Compatible With the BSSN Formulation of the Einstein Equations
• Generating Initial Data in General Relativity using Adaptive Finite Element Methods
• Analytical solution of Brill waves