# Coupling Stokes-Darcy flow with transport on irregular geometries

song, pu (2017) Coupling Stokes-Darcy flow with transport on irregular geometries. Doctoral Dissertation, University of Pittsburgh. (Unpublished)

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## Abstract

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 which can handle irregular geometry.

First, we will use a multiscale mortar finite element method to discretize coupled Stokes-Darcy flows on irregular domains. Especially, we will utilize a special discretization method called multi-point flux mixed finite element method to handle Darcy flow. This method is accurate for rough grids and rough full tensor coefficients, and reduces to a cell-centered pressure scheme. On quadrilaterals and hexahedra the method can be formulated either on the physical space or on the reference space, leading to a non-symmetric or symmetric scheme, respectively. While Stokes region is discretized by standard inf-sup stable elements. The mortar space can be coarser and it is used to approximate the normal stress on the interface and to impose weakly continuity of normal flux. The interfaces can be curved and matching conditions are imposed via appropriate mappings from physical grids to reference grids with flat interfaces.

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.

To discretize the transport equation we develop a local discontinuous Galerkin mortar method. In the method, the subdomain grids need not match and the mortar grid may be much coarser, giving a two-scale method. We weakly impose the boundary condition on the inflow part of the interface and the Dirichlet boundary condition on the elliptic part of the interface via Lagrange multipliers. We develop stability for the concentration and the diffusive flux in the transport equation.

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## Details

Item Type: University of Pittsburgh ETD
Status: Unpublished
Creators/Authors:
song, pupus6@pitt.edupus6
ETD Committee:
Committee ChairYotov, Ivanyotov@math.pitt.edu
Committee MemberLayton, williamwjl@pitt.edu
Committee MemberNeilan, Michaelneilan@pitt.edu
Committee MemberZunino, Paolopaolo.zunino@polimi.it
Date: 28 September 2017
Date Type: Publication
Defense Date: 1 July 2017
Approval Date: 28 September 2017
Submission Date: 20 July 2017
Access Restriction: No restriction; Release the ETD for access worldwide immediately.
Number of Pages: 125
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: Stokes-Darcy flows, mortar finite element, curved interface, multiscale finite element, multipoint flux approximation
Date Deposited: 29 Sep 2017 00:46