Error estimates for a numerical approximation to the compressible barotropic Navier-Stokes equations

TL;DR

This paper introduces a general method using relative energy to derive unconditional error estimates for numerical solutions of the barotropic Navier-Stokes equations, applicable to specific DG/finite element schemes.

Contribution

It develops a novel approach based on relative energy to obtain error bounds that are unconditional, extending previous convergence results for particular schemes.

Findings

Provides an unconditional error estimate for a DG/finite element scheme

Extends previous convergence results to error bounds

Introduces a general methodology applicable to various discretizations

Abstract

We present here a general method based on the investigation of the relative energy of the system, that provides an unconditional error estimate for the approximate solution of the barotropic Navier Stokes equations obtained by time and space discretization. We use this methodology to derive an error estimate for a specific DG/finite element scheme for which the convergence was proved in [27]. This is an extended version of the paper submitted to IMAJNA.