| dc.description.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 with the conforming scheme, the nonconforming scheme is preferred in parallel computing. Since the nonconforming shape functions have local support in most two cells, the sparse matrix stencil will be smaller. The communication across MPI processes is minimal, resulting in better scalability.
Further, the GLPS finite element method for Darcy equations is developed and analyzed. Here, we propose a mixed finite element formulation with the GLPS technique for the Darcy equations, which avoids H(div,Ω) formulation. The equal-order interpolation spaces (P1/P1) are used to approximate the velocity and pressure approximation. In particular, the use of piecewise linear finite elements for both the velocities and the pressure results in ill-posed discretizations. Therefore, the GLPS is proposed in this work to suppress the oscillations in the approximations. Moreover, the boundary conditions are not used strongly in discrete space; hence, the discrete formulation combines standard Galerkin formulations, stabilization terms, and weakly imposed boundary conditions. The proposed bilinear form satisfies an inf-sup condition with respect to generalized local projection stabilized norm, which leads to the well-posedness of the discrete problem. Moreover, the optimal order of convergence is observed with respect to the GLPS norm. Furthermore, the above approach has also been used to study the Stokes problem.
Finally, a priori analysis based on the GLPS Crouzeix-Raviart finite element approximation for the solution of Darcy equations is presented. In the present analysis, two variational stabilization formulations are considered for Darcy equations. The first includes Crouzeix-Raviart (CR) finite element space for the velocity and piecewise constant (P0) polynomial space for the pressure, i.e., (CR/P0), whereas the second includes (CR/CR). The Crouzeix-Raviart space and piecewise constant polynomial space (CR/P0) are an inf-sup stable pair. However, it is known that the finite element pair (CR/P0) does not converge when applied to the Darcy problem. In this work, this convergence issue is managed by GLPS. In the second variant, the pressure is also approximated using the linear nonconforming finite element; that is, CR is used for both velocity and pressure. This equal order finite element pair does not satisfy the inf-sup compatibility, and the GLPS handles the inf-sup violation. Moreover, a first-order convergence is observed for the piecewise constant approximation and 1.5 for the CR finite element approximation of pressure. Finally, the validation of the proposed stabilization schemes is demonstrated with appropriate numerical examples. | en_US |
