13 research outputs found
Error analysis of energy-conservative BDF2-FE scheme for the 2D Navier-Stokes equations with variable density
In this paper, we present an error estimate of a second-order linearized
finite element (FE) method for the 2D Navier-Stokes equations with variable
density. In order to get error estimates, we first introduce an equivalent form
of the original system. Later, we propose a general BDF2-FE method for solving
this equivalent form, where the Taylor-Hood FE space is used for discretizing
the Navier-Stokes equations and conforming FE space is used for discretizing
density equation. We show that our scheme ensures discrete energy dissipation.
Under the assumption of sufficient smoothness of strong solutions, an error
estimate is presented for our numerical scheme for variable density
incompressible flow in two dimensions. Finally, some numerical examples are
provided to confirm our theoretical results.Comment: 22 pages, 1 figure
Modelling of advection-dominated transport in fluid-saturated porous media
The modelling of contaminant transport in porous media is an important topic to geosciences and geo-environmental engineering. An accurate assessment of the spatial and temporal distribution of a contaminant is an important step in the environmental decision-making process. Contaminant transport in porous media usually involves complex non-linear processes that result from the interaction of the migrating chemical species with the geological medium. The study of practical problems in contaminant transport therefore usually requires the development of computational procedures that can accurately examine the non-linear coupling processes involved. However, the computational modelling of the advection-dominated transport process is particularly sensitive to situations where the concentration profiles can exhibit high gradients and/or discontinuities. This thesis focuses on the development of an accurate computational methodology that can examine the contaminant transport problem in porous media where the advective process dominates.The development of the computational method for the advection-dominated transport problem is based on a Fourier analysis on stabilized semi-discrete Eulerian finite element methods for the advection equation. The Fourier analysis shows that under the Courant number condition of Cr=1, certain stabilized finite element scheme can give an oscillation-free and non-diffusive solution for the advection equation. Based on this observation, a time-adaptive scheme is developed for the accurate solution of the one-dimensional advection-dominated transport problem with the transient flow velocity. The time-adaptive scheme is validated with an experimental modelling of the advection-dominated transport problem involving the migration of a chemical solution in a porous column. A colour visualization-based image processing method is developed in the experimental modelling to quantitatively determinate the chemical concentration on the porous column in a non-invasive way. A mesh-refining adaptive scheme is developed for the optimal solution of the multi-dimensional advective transport problem with a time- and space-dependent flow field. Such mesh-refining adaptive procedure is quantitative in the sense that the size of the refined mesh is determined by the Courant number criterion. Finally, the thesis also presents a brief study of a numerical model that is capable to capture coupling Hydro-Mechanical-Chemical processes during the advection-dominated transport of a contaminant in a porous medium
The simulation of single phase, compressible fluid flow in fractured petroleum reservoirs using finite elements
Summary in English.Bibliography: leaves 181-193.In this thesis, commonly used equations governing the flow of fluids are reviewed, from first principles where appropriate. The assumptions that are made in the process are critically assessed and their limitations are discussed. The equations deal with flow through a porous and permeable medium, a single fracture, a network of fractures, and with the coupling of the fracture network and blocks of matrix material
Coupled space-time discontinuous Galerkin method for dynamical modeling in porous media
This thesis deals with coupled space-time discontinuous Galerkin methods for the modeling of dynamical phenomena in fluid saturated porous media. The numerical scheme consists of finite element discretizations in the spatial and in the temporal domain simultaneously. In particular, two major classes of approaches have been investigated.
The first one is the so-called time-discontinuous Galerkin (DGT) method, consisting of discontinuous polynomials in the temporal domain but continuous ones in space. A natural upwind flux treatment is introduced to enforce the continuity condition at discrete time levels. The proposed numerical approach is suitable for solving first-order time-dependend equations. For the second-order equations, an Embedded Velocity Integration (EVI) technique is developed to degenerate a second-order equation into a first-order one. The resulting first-order differential equation with the primary variable in rate term (velocity) can in turn be solved by the time-discontinuous Galerkin method efficiently. Applications concerning both the first- and second-order differential equations as well as wave propagation problems in porous materials are investigated.
The other one is the coupled space-time discontinuous Galerkin (DGST) method, in which neither the spatial nor the temporal approximations pocesses strong continuity. Spatial fluxes combined with flux-weighted constraints are employed to enforce the interelement consistency in space, while the consistency in the time domain is enforced by the temporal upwind flux investigated in the DGT method. As there exists no coupling between the spatial and temporal fluxes, various flux treatments in space and in time are employed independently. The resulting numerical scheme is able to capture the steep gradients or even discontinuities. Applications concerning the single-phase flow within the porous media are presented.Im Rahmen dieser Arbeit werden gekoppelte Raum-Zeit Finite-Element-Methoden fĂŒr die Simulation dynamischer Effekte in fluid-gesĂ€ttigten porösen Materialien entwickelt und numerisch umgesetzt. Dazu wird eine gekoppelte Diskretisierung des rĂ€umlichen und zeitlichen Gebietes vorgenommen. Insbesondere werden zwei Klassen von Verfahren untersucht.
Die erste Methode ist ein sogenanntes zeitlich-diskontinuierliches Galerkin Verfahren (DGT-Methode). Hierbei werden diskontinuierliche AnsĂ€tze in der Zeit und kontinuierliche AnsĂ€tze im Raum verwendet. Die KontinuitĂ€tsnebenbedingung in der Zeit wird durch einen upwind-Flussterm erzwungen. Der Flussterm unterliegt mathematischen Restriktionen und daher ist das resultierende Finite Element Verfahren nur fĂŒr Gleichungen mit zeitlichen Ableitungen erster Ordnung geeignet. Um auch Gleichungen zweiter Ordnungen mit dem entwickelten DGT-Verfahren behandeln zu können, ist die EVI-Methode (Embedded Velocity Integration method) entwickelt worden. Im Rahmen der EVI-Technik wird die Geschwindigkeit als PrimĂ€rvariable gewĂ€hlt und im Bezug auf die gewĂ€hlten zeitlichen AnsĂ€tze integriert. Die auf der Geschwindigkeit basierenden schwachen Formen können wiederum mit der DGT-Methode gelöst werden. Die entwickelten numerischen Raum-Zeit Finite-Elemente-Methoden werden sowohl fĂŒr elastische Wellenausbreitungsprobleme als auch fĂŒr gekoppelte Fragestellungen in porösen Medien angewendet.
AbschlieĂend wird ein rĂ€umlich diskontinuierliches Finite-Element-Verfahren entwickelt und mit den bereits entwickelten zeitlich-diskontinuierlichen Methoden gekoppelt. Die rĂ€umliche KontinuitĂ€tsbedingung wird durch die Entwicklung eines speziellen Flusstermes erzwungen. Es wird gezeigt, dass sich das Verfahren mit den bereits entwickelten Flusstermen fĂŒr die zeitliche KontinuitĂ€t koppeln lĂ€sst. Dies wird durch die Entkopplung der rĂ€umlichen und zeitlichen Flussterme möglich. Das resultierende Raum-Zeit diskontinuierliche Finite-Element-Verfahren wird wiederum auf Strömungsprobleme mischbarer Fluide in porösen Medien angewendet und mit klassischen Methoden verglichen
Time-dependent Stokes-Darcy Flow with Deposition
This thesis investigates two nonlinear systems of time-dependent partial differential equations that model a filtration process. Existence and uniqueness results for the governing equations is established. For each system, a finite element scheme capable of approximating the solutions is investigated. Accompanying numerical experiments corroborate the analytical findings. Finally, an optimization application concerning the design of a filter is discussed and supported with a numerical study