Efficient discretization and preconditioning of the singularly perturbed reaction-diffusion problem

Results on a Mixed Finite Element Approach for a Model Convection-Diffusion Problem

“…The reformulation allows for a new error analysis using an optimal test norm, see e.g. [6,8,9], and for comparison with the known stream-line diffusion (SD) method of discretization that is reviewed in the next section.…”

“…In this paper we analyze mixed variational discretizations of the model convection diffusion problem (1.2), based on the concept of optimal trial norms at the continuous and the discrete levels. The concept of optimal trial norm was developed and used before in e.g., [3,4,5,18,19,21,23,26]. In this paper, for certain discrete test spaces, we find new representations of such norms that allow for sharp error estimates and new analysis for saddle point or mixed variational formulations.…”

“…The focus is on analysis of the variational problem that is written in a mixed formulation and ledas to new stability and approximation results. To improve the rate of convergence in particular norms, we will use the concept of optimal norm, see e.g., [17,19,21,23,22,26,28,3,4], that provides ε-independent stability. In addition, we will take advantage of the mixed reformulations of the variational problem given by the Saddle Point Least Squares (SPLS) method, as presented in [5,6,7,9].…”

Connections between finite difference and finite element approximations

“…For studying stability error estimates and connection with FD methods, we use the concept of optimal trial norm, as presented in [1,2,9,10,12,13,17]. We write the finite difference and the finite element systems for uniformly distributed nodes such that the two corresponding linear systems have the same stiffness matrices, and compare the Right Hand Side (RHS) load vectors for the two methods.…”