Error estimates of the dynamically regularized Lagrange multiplier finite element scheme for the incompressible Navier–Stokes equations

This paper presents fully-discrete error analysis of the dynamically regularized Lagrange multiplier method (DRLM), a novel stabilization approach for the incompressible Navier–Stokes equations that incorporates kinetic energy evolution with a squared Lagrange multiplier regularization term into the system. The first-order implicit-explicit temporal discretization is employed for the DRLM formulation, in combination with the Taylor–Hood finite elements for spatial approximation. We establish optimal H 1 -norm error estimate for the velocity and L 2 -norm error estimate for the pressure through a mathematical induction process. Specifically, we first derive suboptimal error estimates, which serves as a basis for bounding the numerical solutions via the inverse inequalities, and optimal convergence can then be obtained by repeating the similar arguments provided with the uniformly bounded numerical solutions. Furthermore, we prove optimal L 2 -norm error estimate for the velocity by using the negative norm techniques. Finally, numerical experiments are given to verify the theoretical convergence rates.