Convergence analysis of the dynamically regularized Lagrange multiplier method for the incompressible Navier-Stokes equations

TL;DR

This paper provides a convergence analysis of the DRLM method for incompressible Navier-Stokes equations, demonstrating optimal error estimates and confirming theoretical results through numerical experiments.

Contribution

It introduces a convergence analysis for the DRLM method, including optimal error estimates and validation via numerical results.

Findings

Optimal error estimates for velocity and pressure

Numerical results confirm theoretical convergence rates

Unique solvability of the DRLM scheme

Abstract

This paper is concerned with temporal convergence analysis of the recently introduced Dynamically Regularized Lagrange Multiplier (DRLM) method for the incompressible Navier-Stokes equations. A key feature of the DRLM approach is the incorporation of the kinetic energy evolution through a quadratic dynamic equation involving a time-dependent Lagrange multiplier and a regularization parameter. We apply the backward Euler method with an explicit treatment of the nonlinear convection term and show the unique solvability of the resulting first-order DRLM scheme. Optimal error estimates for the velocity and pressure are established based on a uniform bound on the Lagrange multiplier and mathematical induction. Numerical results confirm the theoretical convergence rates and error bounds that decay with respect to the regularization parameter.