Local Projection Stabilization Methods for the Oseen Problem

Abstract

Finite element approximation of fluid flow problems with dominant convection exhibit spurious oscillations. To eliminate these nonphysical oscillations one needs to incorporate stabilizations that can curb the effect of convection. The main aim of this thesis is to design and analyse local projection stabilization based finite element schemes for the Oseen problem.

In chapter \ref{intro}, we have established a background for the Oseen problem citing its main difficulties and a literature survey. In the thesis, we have predominantly discussed the use of three different finite elements methods, namely, the non-conforming Crouzeix-Raviart (${\rm CR}$) method, the $H(\Hdiv;\Omega)$ conforming Raviart-Thomas (${\rm RT}$) element method and the hybrid high order method. The thesis is divided into four chapters.

Chapter \ref{chap1} analyses the edge patchwise local projection (EPLP) stabilized nonconforming finite element methods for the Oseen problem. For approximating the velocity, the lowest-order Crouzeix-Raviart (CR) nonconforming finite element space is considered, whereas for approximating the pressure, two separate discrete spaces are considered, namely, the piecewise constant polynomial space and the lowest-order CR finite element space. The proposed discrete weak formulations are a combination of the standard Galerkin method, EPLP stabilization and weakly imposed boundary condition (Nitsche's technique). We present stability results for both schemes and provide convergence analysis.

{\it A~posteriori} error analysis of the edge patch-wise local projection (EPLP) stabilized Crouzeix-Raviart finite element method is developed in chapter \ref{chap2}. The {\it a~posteriori} analysis is based on the approach of Verf\"urth \cite{verfurth_dual_main}. We prove a stability result for the Oseen equation under a dual norm. The stability result gives an equivalence of error and residual which is independent of the discrete formulation. This gives the freedom of using other stabilizations and finite element spaces in the setting of our analysis. Equivalence of error and residual is exploited to formulate an error estimator which is proven to be reliable. Efficiency estimates show a dependence on the diffusion coefficient.

In chapter \ref{chap3}, we define a Local projection stabilization (LPS) scheme with the Raviart-Thomas( ${\rm RT}_k$) elements for the oseen problem. We show that a divergence free, pressure robust LPS scheme can be designed with ${\rm RT}_k$ elements of order $k \geq 1$. We also show that stability under the streamline upwind Petrov-Galerkin (SUPG) norm can be achieved if the ${\rm RT}_k$ space is enriched with tangential bubbles. The enriched scheme also gives divergence free velocity. We present {\it a~priori} error estimates for both the schemes.

Chapter \ref{chap4} deals with the use of a local projection stabilized Hybrid High-Order scheme for the Oseen problem. We prove an existence-uniqueness result under a SUPG like norm. We derive an optimal order {\it a~priori} error estimate under this norm for equal order polynomial discretization of velocity and pressure spaces.

In the last chapter we provide some concluding remarks on the results proved in the thesis and discuss some future problems to work on.