Generalized local projection stabilized finite element methods for boundary value problems

Abstract

The primary goal of this thesis work is to study a priori analysis based on the generalized local projection stabilized (GLPS) finite element methods for the system of linear partial differential equations of first and second-order, such as the advection-reaction equation, the Darcy equations, and the Stokes problems. It is well-known that applying the standard Galerkin finite element method (FEM) to these types of problems induces spurious oscillations in the numerical solution. Nevertheless, the stability and accuracy of the standard Galerkin solution can be enhanced by applying stabilization techniques. Some of the well-known stabilization techniques are the streamline upwind Petrov-Galerkin methods (SUPG), least-squares methods, residual-free bubbles, continuous interior penalty, subgrid viscosity, and local projection stabilization. The main contribution is to introduce and develop a generalized local projection stabilization for the advection-reaction equation and Darcy equations. Initially, we study the generalized local projection stabilization scheme with conforming and nonconforming finite element spaces for an advection-reaction equation. GLPS technique allows the use of projection spaces on overlapping sets and avoids using a two-level mesh or enrichment of finite element space. Since the Laplacian term is missing in the advection-reaction equation, a different approach is used to derive the coercivity with a stronger local projection streamline derivative (LPSD) norm. An important feature of this LPSD norm is that it provides control with respect to streamline derivatives. Note that the LPSD norm is equivalent to the SUPG norm for an appropriate choice of mesh-dependent parameter. Furthermore, weighted edge integrals of the jumps and the averages of the discrete solution at the interfaces need to be added to the nonconforming bilinear form to derive the stability and error estimates for the nonconforming discrete formulation. Though the analysis of nonconforming GLPS is challenging in comparison...