Convergence in the incompressible limit of finite element approximations based on the Hu-Washizu formulation

2006 ◽  
Vol 104 (2) ◽  
pp. 151-175 ◽  
Author(s):  
B. P. Lamichhane ◽  
B. D. Reddy ◽  
B. I. Wohlmuth
Keyword(s):  
Finite Element ◽  
Incompressible Limit ◽  
Finite Element Approximations

scholarly journals Weakly Imposed Symmetry and Robust Preconditioners for Biot’s Consolidation Model

2017 ◽  
Vol 17 (3) ◽  
pp. 377-396 ◽  
Author(s):  
Trygve Bærland ◽  
Jeonghun J. Lee ◽  
Kent-Andre Mardal ◽  
Ragnar Winther
Keyword(s):  
Finite Element ◽  
Finite Element Methods ◽  
Linear Elasticity ◽  
Discrete Systems ◽  
Model Parameters ◽  
Incompressible Limit ◽  
Biot Model ◽  
Finite Element Approximations ◽  
Consolidation Model ◽  
Biot’S Consolidation

AbstractWe discuss the construction of robust preconditioners for finite element approximations of Biot’s consolidation model in poroelasticity. More precisely, we study finite element methods based on generalizations of the Hellinger–Reissner principle of linear elasticity, where the stress tensor is one of the unknowns. The Biot model has a number of applications in science, medicine, and engineering. A challenge in many of these applications is that the model parameters range over several orders of magnitude. Therefore, discretization procedures which are well behaved with respect to such variations are needed. The focus of the present paper will be on the construction of preconditioners, such that the preconditioned discrete systems are well-conditioned with respect to variations of the model parameters as well as refinements of the discretization. As a byproduct, we also obtain preconditioners for linear elasticity that are robust in the incompressible limit.

scholarly journals 3D Unsteady Diffusion and Reaction-Diffusion with Singularities by GFEM with 27-Node Hexahedrons

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Estaner Claro Romão
Keyword(s):  
Finite Element ◽  
Variable Temperature ◽  
Three Dimensional ◽  
Reaction Diffusion ◽  
Third Order ◽  
Order Of Accuracy ◽  
Finite Element Approximations ◽  
Temperature Solution ◽  
The One ◽  
Primary Variable

The Galerkin Finite Element Method (GFEM) with 8- and 27-node hexahedrons elements is used for solving diffusion and transient three-dimensional reaction-diffusion with singularities. Besides analyzing the results from the primary variable (temperature), the finite element approximations were used to find the derivative of the temperature in all three directions. This technique does not provide an order of accuracy compatible with the one found in the temperature solution; thereto, a calculation from the third order finite differences is proposed here, which provide the best results, as demonstrated by the first two applications proposed in this paper. Lastly, the presentation and the discussion of a real application with two cases of boundary conditions with singularities are proposed.

scholarly journals On the Adaptive Selection of the Parameter in Stabilized Finite Element Approximations

2013 ◽  
Vol 51 (3) ◽  
pp. 1585-1609 ◽  
Author(s):  
Mark Ainsworth ◽  
Alejandro Allendes ◽  
Gabriel R. Barrenechea ◽  
Richard Rankin
Keyword(s):  
Finite Element ◽  
Adaptive Selection ◽  
Stabilized Finite Element ◽  
Finite Element Approximations ◽  
Selection Of

Two-scale sparse finite element approximations

2015 ◽  
Vol 59 (4) ◽  
pp. 789-808
Author(s):  
Fang Liu ◽  
JinWei Zhu
Keyword(s):  
Finite Element ◽  
Finite Element Approximations
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