A Generalized Weak Galerkin method for Oseen equation

TL;DR

This paper introduces a generalized weak Galerkin finite element method for the time-dependent Oseen equation, providing new approximation frameworks, stability analysis, and optimal error estimates validated through numerical experiments.

Contribution

It develops a novel generalized weak Galerkin framework for the Oseen equation, including new stability conditions and convergence analysis, with validated optimal error estimates.

Findings

Established convergence and stability of the method.

Derived optimal order error estimates.

Validated results with benchmark examples.

Abstract

In this work, the authors introduce a generalized weak Galerkin (gWG) finite element method for the time-dependent Oseen equation. The generalized weak Galerkin method is based on a new framework for approximating the gradient operator. Both a semi-discrete and a fully-discrete numerical scheme are developed and analyzed for their convergence, stability, and error estimates. A generalized {\em{inf-sup}} condition is developed to assist the convergence analysis. The backward Euler discretization is employed in the design of the fully-discrete scheme. Error estimates of optimal order are established mathematically, and they are validated numerically with some benchmark examples.