Adaptive Finite Element Methods for Flows in Porous Media
Lead Research Organisation:
University of Nottingham
Department Name: Sch of Mathematical Sciences
Abstract
In this thesis we consider the numerical approximation of porous media flows employing finite element methods. In particular, we shall derive efficient and reliable a posteriori error bounds for the computation of certain output functionals of practical relevance. These error estimates will then be implemented within automatic adaptive mesh refinement algorithms in order to compute solutions to within a prescribed error tolerance. Finally, we shall consider the application of these techniques to problems of engineering interest, such as, for example, a problem based on the geological units found at the Sellafield site in the UK.
Organisations
People |
ORCID iD |
| Paul Houston (Primary Supervisor) | |
| Connor Rourke (Student) |
Studentship Projects
| Project Reference | Relationship | Related To | Start | End | Student Name |
|---|---|---|---|---|---|
| EP/N50970X/1 | 01/10/2016 | 30/09/2021 | |||
| 2281583 | Studentship | EP/N50970X/1 | 01/10/2019 | 28/02/2023 | Connor Rourke |
| EP/R513283/1 | 01/10/2018 | 30/09/2023 | |||
| 2281583 | Studentship | EP/R513283/1 | 01/10/2019 | 28/02/2023 | Connor Rourke |
| Description | Modelling the earth as a saturated porous media, the flow of groundwater is governed by Darcy's law and conservation of mass. A mixed finite element method can be used to obtain a numerical approximation of the Darcy velocity and the pressure of the flow. Often, the accuracy of these solutions is not important, but more so the accuracy of certain quantities involving these solutions, referred to as quantities of interest. A mesh underlies the numerical approximation of the Darcy velocity and pressure; work in a posteriori error estimation seeks to refine this mesh only where necessary, to obtain an approximation of the quantity of interest to within some user-defined tolerance. The resulting procedures are referred to as (goal-oriented) adaptive finite element methods. Moreover, using a so-called adjoint problem, linked directly to the quantity of interest, one may use dual-weighted-residual (DWR) error estimation to estimate numerical errors in the approximation of the quantity of interest, and thus select particular elements in the mesh for refinement. In this award, general quantities of interest, taking the form of linear functionals of the solutions were considered, employing both Raviart-Thomas (RT) and Brezzi-Douglas-Marini (BDM) finite elements for the Darcy velocity. Furthermore, an adaptive finite element method was implemented using DWR error estimation. Numerical examples investigated within this award gave empirical evidence that using RT elements for the Darcy velocity results in a sub-optimal rate of convergence for these quantities of interest. However, the main focus of this award has been the so-called travel time functional. This quantity refers to the time-taken for a particle trajectory to travel from a given initial position to the boundary of the problem domain; such is a simple model to represent leaked solutes of radioactive waste that has been buried deep underground. In 2013, this problem was investigated by Joseph Collis in his thesis "Error Estimation for PDEs with Random Inputs" in which the DWR error estimation for the travel time functional was achieved via a generalised Taylor series expansion of the derivative of the travel time functional. In my award, a novel re-understanding, and re-implementation of this adaptive finite element method has been realised. An exact expression for the derivative of the travel time functional to be used in DWR error estimation has been established. This novel result and the subsequent error estimation for the travel time functional has been applied to a variety of different numerical examples, including an example based on the Sellafield site, in the UK, as also implemented by Collis in his thesis. The resulting data showed an excellent performance of the error estimate utilising the exact expression for the derivative of the travel time functional. The above work on the travel time functional has been written up as the paper: "Linearisation of the Travel Time Functional in Porous Media Flows" which has been accepted, and is currently in press at the time of writing, to be published in SIAM Journal on Scientific Computing. |
| Exploitation Route | Firstly, the problem with an RT approximation for the Darcy velocity seems to be an open problem still. Under DWR error estimation, rigorously proving why this sub-optimality appears needs to be addressed, as well as whether under a different construction of the error estimate, this "lost" convergence can be retrieved. With regards to the travel time functional, this functional is an example of a nonlinear, and possibly unbounded, quantity of interest. The framework surrounding the derivation of the exact expression of its derivative may be utilised and applied to other, similar, quantities of interest. Furthermore, it is worth noting that the work on the travel time functional is independent of where the velocity field driving the particle trajectory comes from, and thus the travel time functional could be considered in other flow models, for example in a Stokes flow or a more general Navier-Stokes flow. Within my award, extension to consider flow and travel time in fractured porous media is currently being undertaken, for example. |
| Sectors | Aerospace, Defence and Marine,Agriculture, Food and Drink,Energy,Environment,Government, Democracy and Justice,Other |
| URL | https://arxiv.org/abs/2111.15504 |