Convergence analysis of dynamically regularized Lagrange multiplier pressure correction method for the incompressible Navier-Stokes equations
This work addresses numerical stability and efficiency in fluid dynamics simulations, but it is incremental as it builds on existing DRLM methods with a focus on error analysis.
The authors tackled the problem of solving the incompressible Navier-Stokes equations by proposing a first-order pressure-correction scheme using Dynamically Regularized Lagrange Multiplier methods, resulting in a fully decoupled algorithm with unconditional energy stability and optimal error estimates for velocity and pressure, validated through numerical experiments.
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.