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Domain Decomposition And Time-Splitting Methods For The Biot System Of Poroelasticity

Jayadharan, Manu (2021) Domain Decomposition And Time-Splitting Methods For The Biot System Of Poroelasticity. Doctoral Dissertation, University of Pittsburgh. (Unpublished)

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In this thesis, we develop efficient mixed finite element methods to solve the Biot system of poroelasticity, which models the flow of a viscous fluid through a porous medium along with the deformation of the medium. We study non-overlapping domain decomposition techniques and sequential splitting methods to reduce the computational complexity of the problem. The solid deformation is
modeled with a mixed three-field formulation with weak stress
symmetry. The fluid flow is modeled with a mixed Darcy formulation.

We introduce displacement and pressure Lagrange multipliers on the
subdomain interfaces to impose weakly the continuity of normal stress and
normal velocity, respectively. The global problem is reduced to an
interface problem for the Lagrange multipliers, which is solved by a
Krylov space iterative method. We study both monolithic and split
methods. For the monolithic method, the cases of matching and non-matching subdomain grid interfaces are analyzed separately. For both cases, a coupled displacement-pressure
interface problem is solved, with each iteration requiring the
solution of local Biot problems. For the case of matching subdomain grids, we show that the resulting interface
operator is positive definite and analyze the convergence of the
iteration. For the non-matching subdomain grid case, we use a multiscale mortar mixed finite element (MMMFE) approach.

We further study drained split and fixed stress Biot
splittings, in which case we solve separate interface problems
requiring elasticity and Darcy solves. We analyze the
stability of the split formulations. We also use numerical experiments to
illustrate the convergence of the domain decomposition
methods and compare their accuracy and efficiency in the monolithic and time-splitting settings.

Finally, we present a novel space-time domain decomposition technique for the mixed finite element formulation of a parabolic equation. This method is motivated by the MMMFE method, where we split the space-time domain into multiple subdomains with space-time grids of different sizes. Scalar Lagrange multiplier (mortar) functions are introduced to enforce weakly the continuity of the normal component of the mixed finite element flux variable over the space-time interfaces. We analyze the new method and numerical experiments are developed to illustrate and confirm the theoretical results.


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Item Type: University of Pittsburgh ETD
Status: Unpublished
CreatorsEmailPitt UsernameORCID
ETD Committee:
TitleMemberEmail AddressPitt UsernameORCID
Committee ChairYotov,
Committee MemberNeilan,
Committee MemberTrenchea,
Committee MemberVohralik,
Date: 8 October 2021
Date Type: Publication
Defense Date: 22 July 2021
Approval Date: 8 October 2021
Submission Date: 24 July 2021
Access Restriction: No restriction; Release the ETD for access worldwide immediately.
Number of Pages: 193
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: Efficient CFD solvers
Date Deposited: 08 Oct 2021 19:46
Last Modified: 08 Oct 2021 19:46


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