Adaptive Finite Elements and Colliding Black Holes

TL;DR

This paper discusses the development of an adaptive finite element code for simulating colliding black holes, addressing complex mathematical and computational challenges in solving Einstein's equations.

Contribution

It introduces a novel finite element-based adaptive mesh method for initial data problems in black hole collision simulations, enhancing computational accuracy and efficiency.

Findings

Developed an adaptive tetrahedral mesh generation technique.

Implemented a multigrid solver for nonlinear elliptic equations.

Addressed key mathematical challenges in simulating black hole collisions.

Abstract

According to the theory of general relativity, the relative acceleration of masses generates gravitational radiation. Although gravitational radiation has not yet been detected, it is believed that extremely violent cosmic events, such as the collision of black holes, should generate gravity waves of sufficient amplitude to detect on earth. The massive Laser Interferometer Gravitational-wave Observatory, or LIGO, is now being constructed to detect gravity waves. Consequently there is great interest in the computer simulation of black hole collisions and similar events, based on the numerical solution of the Einstein field equations. In this note we introduce the scientific, mathematical, and computational problems and discuss the development of a computer code to solve the initial data problem for colliding black holes, a nonlinear elliptic boundary value problem posed in an unbounded three dimensional domain which is a key step in solving the full field equations. The code is based on finite elements, adaptive meshes, and a multigrid solution process. Here we will particularly emphasize the mathematical and algorithmic issues arising in the generation of adaptive tetrahedral meshes.