A pressure-stabilized projection Lagrange--Galerkin scheme for the transient Oseen problem

TL;DR

This paper introduces a pressure-stabilized projection Lagrange--Galerkin scheme for the transient Oseen problem, offering computational efficiency, unconditional stability, and improved accuracy for small viscosity through equal-order approximation and pressure stabilization.

Contribution

It develops a novel pressure-stabilized scheme that allows for equal-order velocity-pressure approximation with enhanced stability and accuracy for small viscosity in transient Oseen problems.

Findings

The scheme achieves higher accuracy for small viscosity.

Error estimates are established for velocity and pressure.

Numerical tests confirm the scheme's efficiency and stability.

Abstract

We propose and analyze a pressure-stabilized projection Lagrange--Galerkin scheme for the transient Oseen problem. The proposed scheme inherits the following advantages from the projection Lagrange--Galerkin scheme. The first advantage is computational efficiency. The scheme decouples the computation of each component of the velocity and pressure. The other advantage is essential unconditional stability. Here we also use the equal-order approximation for the velocity and pressure, and add a symmetric pressure stabilization term. This enriched pressure space enables us to obtain accurate solutions for small viscosity. First, we show an error estimate for the velocity for small viscosity. Then we show convergence results for the pressure. Numerical examples of a test problem show higher accuracy of the proposed scheme for small viscosity.