Convergence study of IB methods for Stokes equations with non-periodic boundary conditions

08/15/2021
by   Zhilin Li, et al.
0

Peskin's Immersed Boundary (IB) model and method are among the most popular modeling tools and numerical methods. The IB method has been known to be first order accurate in the velocity. However, almost no rigorous theoretical proof can be found in the literature for Stokes equations with a prescribed velocity boundary condition. In this paper, it has been shown that the pressure of the Stokes equation has convergence order O(√(h)) in the L^2 norm while the velocity has O(h) convergence in the infinity norm in two-dimensions (2D). The proofs are based on the idea of the immersed interface method, and the convergence proof of the IB method for elliptic interface problems <cit.>. The proof is intuitive and the conclusion can apply to different boundary conditions as long as the problem is well-posed. The proof process also provides an efficient way to decouple the system into three Helmholtz/Poisson equations without affecting the accuracy. A non-trivial numerical example is also provided to confirm the theoretical analysis.

READ FULL TEXT
research
02/16/2023

Second order convergence of a modified MAC scheme for Stokes interface problems

Stokes flow equations have been implemented successfully in practice for...
research
02/23/2020

High-order Methods for a Pressure Poisson Equation Reformulation of the Navier-Stokes Equations with Electric Boundary Conditions

Pressure Poisson equation (PPE) reformulations of the incompressible Nav...
research
07/31/2022

On the reconstruction of unknown driving forces from low-mode observations in the 2D Navier-Stokes Equations

This article is concerned with the problem of determining an unknown sou...
research
12/22/2022

No pressure? Energy-consistent ROMs for the incompressible Navier-Stokes equations with time-dependent boundary conditions

This work presents a novel reduced-order model (ROM) for the incompressi...
research
09/21/2022

Construction of boundary conditions for Navier-Stokes equations from the moment system

This work concerns with boundary conditions (BCs) of the linearized mome...
research
07/16/2023

On posterior consistency of data assimilation with Gaussian process priors: the 2D Navier-Stokes equations

We consider a non-linear Bayesian data assimilation model for the period...

Please sign up or login with your details

Forgot password? Click here to reset