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

TL;DR

This paper presents a finite element method approach for solving Einstein constraint equations on multi-block domains, demonstrating superconvergence and applying it to generate initial data for relativistic simulations.

Contribution

It introduces a finite element framework for Einstein constraints on multi-block grids, including superconvergence analysis and initial data generation for relativistic simulations.

Findings

Superconvergence at mesh vertices due to local symmetry.

Unstructured mesh refinements do not improve data quality.

Feasibility demonstrated through evolution and gravitational wave extraction.

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.