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

TL;DR

This paper introduces a first-order pressure-correction scheme for the incompressible Navier-Stokes equations that is fully decoupled, energy stable, and rigorously analyzed for optimal error estimates, validated through numerical experiments.

Contribution

It develops a novel, fully decoupled pressure-correction method incorporating DRLM techniques with proven stability and optimal convergence for velocity and pressure.

Findings

The scheme is unconditionally energy stable.

Optimal error estimates are established for velocity and pressure.

Numerical experiments confirm the method's accuracy and robustness.

Abstract

We propose first-order pressure-correction scheme for the incompressible Navier-Stokes equations, incorporating the recently developed the Dynamically Regularized Lagrange Multiplier (DRLM) methods. The resulting algorithms are fully decoupled and require solving only Poisson-type equations at each time step. Moreover, it exhibits unconditional energy stability. This paper provides a rigorous error analysis for the first-order scheme, establishing optimal error estimates for both velocity and pressure. Specifically, we employ mathematical induction to derive sharp velocity error bounds, while leveraging the inf-sup condition to prove optimal convergence rate for the pressure. To validate our theoretical findings, we present two numerical experiments demonstrating the accuracy and robustness of the method.