Saddle point least squares for the reaction–diffusion problem

Efficient discretization and preconditioning of the singularly perturbed Reaction Diffusion problem

“…In the above problem, as mentioned we can replace the action of the discrete operator corresponding to a opt (•, •) by a preconditioner. The following result regarding the stability of these spaces was proved in [6].…”

“…Two different choices the projection type trial space, based on the inner product chosen for Mh , are considered. More details about the construction of these trial spaces can be found in [6].…”

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

“…In the next sections, we will show that the optimal discrete norm and c(h, ε) improve as we consider different test spaces. Numerical tests for the case 1 0 f (x) dx = 0, show that as ε << h, the linear finite element solution of (4.2) presents non-physical oscillations, see [5]. The behavior of the standard linear finite element approximation of (4.2) motivates the use of non-standard discretization approaches, such as the saddle point least square or Petrov-Galerkin methods.…”

“…A saddle point least square (SPLS) approach for solving (4.1) has been used before, for example in [11,21,5]. For V = Q = H 1 0 (0, 1), we look for finding (w, u)…”

“…There is a tremendous amount of literature addressing these types of problems, see e.g. [24,28,30,32,34,21,5,14]. 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.…”