KHATTATOV, ELDAR
(2018)
EFFICIENT DISCRETIZATION TECHNIQUES AND DOMAIN DECOMPOSITION METHODS FOR POROELASTICITY.
Doctoral Dissertation, University of Pittsburgh.
(Unpublished)
Abstract
This thesis develops a new mixed finite element method for linear elasticity model with
weakly enforced symmetry on simplicial and quadrilateral grids. Motivated by the
multipoint flux mixed finite element method (MFMFE) for flow in porous media, the method utilizes
the lowest order Brezzi-Douglas-Marini finite element spaces and the trapezoidal
(vertex) quadrature rule in order to localize the interaction of degrees of freedom. Particularly,
this allows for local elimination of stress and rotation variables around each
vertex and leads to a cell-centered system for the displacements. The stability analysis
shows that the method is well-posed on simplicial and quadrilateral grids. Theoretical
and numerical results indicate first-order convergence for all variables in the natural
norms.
Further discussion of the application of said Multipoint Stress Mixed Finite Element
(MSMFE) method to the Biot system for poroelasticity is then presented. The flow part of the proposed
model is treated in the MFMFE framework, while the mixed formulation for
the elasticity equation is adopted for the use of the MSMFE technique.
The extension of the MFMFE method to an arbitrary order finite volume scheme
for solving elliptic problems on quadrilateral and hexahedral grids that reduce the underlying
mixed finite element method to cell-centered pressure system is also discussed.
A Multiscale Mortar Mixed Finite Element method for the linear elasticity on non-matching
multiblock grids is also studied. A mortar finite element space is introduced
on the nonmatching interfaces. In this mortar space the trace of
the displacement is approximated, and continuity of normal stress is then weakly imposed. The condition
number of the interface system is analyzed and optimal order of convergence is shown
for stress, displacement, and rotation. Moreover, at cell centers, superconvergence is
proven for the displacement variable. Computational results using an efficient parallel
domain decomposition algorithm are presented in confirmation of the theory for all
proposed approaches.
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Details
| Item Type: |
University of Pittsburgh ETD
|
| Status: |
Unpublished |
| Creators/Authors: |
|
| ETD Committee: |
|
| Date: |
28 June 2018 |
| Date Type: |
Publication |
| Defense Date: |
27 March 2018 |
| Approval Date: |
28 June 2018 |
| Submission Date: |
2 April 2018 |
| Access Restriction: |
No restriction; Release the ETD for access worldwide immediately. |
| Number of Pages: |
228 |
| Institution: |
University of Pittsburgh |
| Schools and Programs: |
Dietrich School of Arts and Sciences > Mathematics |
| Degree: |
PhD - Doctor of Philosophy |
| Thesis Type: |
Doctoral Dissertation |
| Refereed: |
Yes |
| Uncontrolled Keywords: |
mixed finite element methods, finite volume schemes, multiscale mortar MFEM, domain decomposition, linear elasticity, Biot consolidation model |
| Date Deposited: |
28 Jun 2018 15:21 |
| Last Modified: |
28 Jun 2018 15:21 |
| URI: |
http://d-scholarship.pitt.edu/id/eprint/34001 |
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