1,654 research outputs found
Discretizations and Solvers for Coupling Stokes-Darcy Flows With Transport
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
An embedded-hybridized discontinuous Galerkin method for the coupled Stokes-Darcy system
We introduce an embedded-hybridized discontinuous Galerkin (EDG-HDG) method
for the coupled Stokes-Darcy system. This EDG-HDG method is a pointwise
mass-conserving discretization resulting in a divergence-conforming velocity
field on the whole domain. In the proposed scheme, coupling between the Stokes
and Darcy domains is achieved naturally through the EDG-HDG facet variables.
\emph{A priori} error analysis shows optimal convergence rates, and that the
velocity error does not depend on the pressure. The error analysis is verified
through numerical examples on unstructured grids for different orders of
polynomial approximation
A compatible embedded-hybridized discontinuous Galerkin method for the Stokes--Darcy-transport problem
We present a stability and error analysis of an embedded-hybridized
discontinuous Galerkin (EDG-HDG) finite element method for coupled
Stokes--Darcy flow and transport. The flow problem, governed by the
Stokes--Darcy equations, is discretized by a recently introduced exactly mass
conserving EDG-HDG method while an embedded discontinuous Galerkin (EDG) method
is used to discretize the transport equation. We show that the coupled flow and
transport discretization is compatible and stable. Furthermore, we show
existence and uniqueness of the semi-discrete transport problem and develop
optimal a priori error estimates. We provide numerical examples illustrating
the theoretical results. In particular, we compare the compatible EDG-HDG
discretization to a discretization of the coupled Stokes--Darcy and transport
problem that is not compatible. We demonstrate that where the incompatible
discretization may result in spurious oscillations in the solution to the
transport problem, the compatible discretization is free of oscillations. An
additional numerical example with realistic parameters is also presented
Simulation of rock salt dissolution and its impact on land subsidence
Extensive land subsidence can occur due to subsurface dissolution of evaporites such as halite and gypsum. This paper explores techniques to simulate the salt dissolution forming an intrastratal karst, which is embedded in a sequence of carbonates, marls, anhydrite and gypsum. A numerical model is developed to simulate laminar flow in a subhorizontal void, which corresponds to an opening intrastratal karst. The numerical model is based on the laminar steady-state Stokes flow equation, and the advection dispersion transport equation coupled with the dissolution equation. The flow equation is solved using the nonconforming Crouzeix-Raviart (CR) finite element approximation for the Stokes equation. For the transport equation, a combination between discontinuous Galerkin method and multipoint flux approximation method is proposed. The numerical effect of the dissolution is considered by using a dynamic mesh variation that increases the size of the mesh based on the amount of dissolved salt. The numerical method is applied to a 2D geological cross section representing a Horst and Graben structure in the Tabular Jura of northwestern Switzerland. The model simulates salt dissolution within the geological section and predicts the amount of vertical dissolution as an indicator of potential subsidence that could occur. Simulation results showed that the highest dissolution amount is observed near the normal fault zones, and, therefore, the highest subsidence rates are expected above normal fault zones
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