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Discretizations and Solvers for Coupling Stokes-Darcy Flows With Transport

Vassilev, Danail Hristov (2010) Discretizations and Solvers for Coupling Stokes-Darcy Flows With Transport. Doctoral Dissertation, University of Pittsburgh. (Unpublished)

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This thesis studies a mathematical model, in which Stokes-Darcy flow system is coupled with a transport equation. The objective is to develop stable and convergent numerical schemes that could be used in environmental applications. Special attention is given to discretization methods that conserve mass locally. First, we present a global saddle point problem approach, which employs the discontinuous Galerkin method to discretize the Stokes equations and the mimetic finite difference method to discretize the Darcy equation. We show how the numerical scheme can be formulated on general polygonal (polyhedral in three dimensions) meshes if suitable operators mapping from degrees of freedom to functional spaces are constructed. The scheme is analyzed and error estimates are derived. A hybridization technique is used to solve the system effectively. We ran several numerical experiments to verify the theoretical convergence rates and depending on the mesh type we observed superconvergence of the computed solution in the Darcy region.Another approach that we use to deal with the flow equations is based on non-overlapping domain decomposition. Domain decomposition enables us to solve the coupled Stokes-Darcy flow problem in parallel by partitioning the computational domain into subdomains, upon which families of coupled local problems of lower complexity are formulated. The coupling of the subdomain problems is removed through an iterative procedure. We investigate the properties of this method and derive estimates for the condition number of the associated algebraic system. Results from computer tests supporting the convergence analysis of the method are provided. To discretize the transport equation we use the local discontinuous Galerkin (LDG) method, which can be thought as a discontinuous mixed finite element method, since it approximates both the concentration and the diffusive flux. We develop stability and convergence analysis for the concentration and the diffusive flux in the transport equation. The numerical error is a combination of the LDG discretization error and the error from the discretization of the Stokes-Darcy velocity. Several examples verifying the theory and illustrating the capabilities of the method are presented.


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Item Type: University of Pittsburgh ETD
Status: Unpublished
CreatorsEmailPitt UsernameORCID
Vassilev, Danail Hristovdhv1@pitt.eduDHV1
ETD Committee:
TitleMemberEmail AddressPitt UsernameORCID
Committee ChairYotov, Ivanyotov@math.pitt.eduYOTOV
Committee MemberTrenchea, Catalintrenchea@pitt.eduTRENCHEA
Committee MemberLipnikov,
Committee MemberSussman, Mikesussmanm@math.pitt.ed
Committee MemberLayton, Williamwjl@pitt.eduWJL
Date: 1 October 2010
Date Type: Completion
Defense Date: 16 July 2010
Approval Date: 1 October 2010
Submission Date: 26 July 2010
Access Restriction: No restriction; Release the ETD for access worldwide immediately.
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: aquifer simulation; boundary value problems; computational fluid dynamics; contaminant transport; discontinuous Galerkin method; domain decomposition; finite element method; groundwater flow; mimetic finite difference method; numerical methods; partial differential equations; stable and convergent methods; Stokes-Darcy flows; surface water flow
Other ID:, etd-07262010-095252
Date Deposited: 10 Nov 2011 19:54
Last Modified: 15 Nov 2016 13:47


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