A Toy Model for Testing Finite Element Methods to Simulate Extreme-Mass-Ratio Binary Systems

TL;DR

This paper introduces a simplified toy model to test adaptive finite element methods for simulating extreme-mass-ratio binary systems, which are key sources of gravitational waves for LISA, addressing the challenge of multiple scales involved.

Contribution

It presents a novel toy model simulation framework to evaluate finite element methods for complex binary systems with large scale disparities.

Findings

Finite element methods can effectively handle multi-scale problems.

Adaptive mesh refinement improves simulation accuracy.

The toy model demonstrates potential for future realistic binary system modeling.

Abstract

Extreme mass ratio binary systems, binaries involving stellar mass objects orbiting massive black holes, are considered to be a primary source of gravitational radiation to be detected by the space-based interferometer LISA. The numerical modelling of these binary systems is extremely challenging because the scales involved expand over several orders of magnitude. One needs to handle large wavelength scales comparable to the size of the massive black hole and, at the same time, to resolve the scales in the vicinity of the small companion where radiation reaction effects play a crucial role. Adaptive finite element methods, in which quantitative control of errors is achieved automatically by finite element mesh adaptivity based on posteriori error estimation, are a natural choice that has great potential for achieving the high level of adaptivity required in these simulations. To demonstrate this, we present the results of simulations of a toy model, consisting of a point-like source orbiting a black hole under the action of a scalar gravitational field.