Thanh Hai Ong

Thanh Hai Ong, Claire E. Heaney, Chang-Kye Lee et al.

We present in this paper a rigorous theoretical framework to show stability, convergence and accuracy of improved edge-based and face-based smoothed finite element methods (bESFEM and bFS-FEM) for nearly-incompressible elasticity problems. The crucial idea is that the space of piecewise linear polynomials used for the displacements is enriched with bubble functions on each element, while the pressure is a piecewise constant function. The meshes of triangular or tetrahedral elements required by these methods can be generated automatically. The enrichment induces a softening in the bilinear form allowing the weakened weak (W2)procedure to produce a high-quality solution, free from locking and that does not oscillate. We prove theoretically that both methods confirm the uniform inf-sup and convergence conditions. Four numerical examples are given to validate the reliability of the bES-FEM and bFS-FEM.

Thanh Hai Ong, Duc Cam Hai Vo, Thi-Thao-Phuong Hoang

The paper is concerned with the derivation and analysis of nonoverlapping domain decomposition for heterogeneous, anisotropic diffusion problems discretized by the finite element cell-centered (FECC) scheme. Differently from the standard finite element method (FEM), the FECC method involves only cell unknowns and satisfies local conservation of fluxes by using a technique of dual mesh and multipoint flux approximations to construct the discrete gradient operator. Consequently, if the domain is decomposed into nonoverlapping subdomains, the transmission conditions (on the interfaces between subdomains) associated with the FECC scheme are different from those of the standard FEM. However, the substructuring procedure as well as the Neumann-Neumann type preconditioner can be adapted to the domain decomposition-based FECC method naturally. Convergence analysis of a preconditioned iterative algorithm, namely the Dirichlet-Neumann to Neumann-Neumann algorithm, associated with the discrete FECC interface problem is the main focus of this work. Two dimensional numerical results for two subdomains with conforming meshes demonstrate that the preconditioned iterative algorithm converges independently of the mesh size and the coefficient jump.