Least-squares finite element method for singularly perturbed problems and the Oseen problem
This dissertation is a summary of the graduate study in the past few years. It contains two parts: the first part is about the least-squares finite element method for the convection-dominated diffusion reaction problems and the second part focuses on the least-squares formulation for the Oseen problem. Convection-dominated diffusion reaction problem is well known at its special property that its solution exhibits boundary/interior layers, which makes the computation extremely difficult. The first part of the dissertation presents three least-squares formulations based on the first-order system (FOSLS) with different ways of imposing boundary conditions. We establish the coercivity for all formulations. To derive the a priori error estimate, we consider a stronger norm, which incorporates the norm of the streamline derivative. The three formulations achieve the same convergence rate, provided that the mesh in the layer regimes is fine enough. Finally, to increase the computation accuracy and reduce the computation cost, the adaptive algorithm is implemented for the numerical tests. The second part of the dissertation studies the least-squares finite element method for the linearized, stationary Navier-Stokes equation based on the stress-velocity-pressure formulation in d dimensions (d = 2 or 3). The least-squares functional is simply defined as the sum of the squares of the L2 norm of the residuals. It is shown that the homogeneous least-squares functional is elliptic and continuous in the H(∇·;Ω)d × H1(Ω)d × L2(Ω) norm. This immediately implies that the a priori error estimate of the conforming least-squares finite element approximation is optimal in the energy norm. The L 2 norm error estimate for the velocity is also established through a refined duality argument. Moreover, when the right-hand side f belongs only to L2(Ω) d, we derive an a priori error bound in a weaker norm, i.e., the L2(Ω)d × d × H 1(Ω)d × L 2(Ω) norm.
Cai, Purdue University.
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