Volume 9, Issue 4

We propose a finite element discretization of a problem issued from the fictitious domain technique. We prove a priori error estimates for the discrete problem and present numerical experiments which are in good agreement with the analysis.

The main goal of this paper is to establish the convergence of mimetic discretizations of the first-order system that describes linear diffusion. Specifically, mimetic discretizations based on the support-operators methodology (SO) have been applied successfully in a number of application areas, including diffusion and electromagnetics. These discretizations have demonstrated excellent robustness, however, a rigorous convergence proof has been lacking. In this research, we prove convergence of the SO discretization for linear diffusion by first developing a connection of this mimetic discretization with Mixed Finite Element (MFE) methods. This connection facilitates the application of existing tools and error estimates from the finite element literature to establish convergence for the SO discretization. The convergence properties of the SO discretization are verified with numerical examples.

In this paper, we consider an overlapping domain decomposition method for second order elliptic boundary value problems. The method is based on nonmatching simplicial triangulations of the subdomains and relies on an appropriate splitting of the variational formulation. The coupling conditions are taken care of by distributed Lagrange multipliers. The LBB condition is established and an efficient preconditioner is suggested.

In this paper, based on a specially chosen domain decomposition, we construct an overlapping additive Schwarz preconditioner according to the framework of [Brenner, Numer. Math. 72: 419–447, 1996] for the Morley element and show that its condition number is optimal; we analyze in details the structure of this preconditioner, and after proper choices of inexact solvers, we obtain a vertex space preconditioner for the Morley element. Compared with the preconditioners constructed in [Huang, J. Comp. Math. 17: 615–628, 1999, Shi and Xie, J. Comp. Math. 16: 289–304, 1998, Xie, Domain Decomposition and Multigrid Methods for Nonconforming Plate Elements, Chinese Academy of Sciences, 1998], this preconditioner has some advantages, i.e., the computational cost adds little, but the condition number improves greatly.

In this note, we extend our studies on finite element Galerkin schemes for elliptic variational inequalities of first to the one of second kind. Especially we perform the corresponding a posteriori error analysis for a model of the flow of a Bingham fluid.