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

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.