Home Articles posted by Zhaonan Dong

Efficient numerical simulation of single-phase flow in three-dimensional fractured porous media

Daniel Zegarra Vasquez: Thursday, 15th May 2025 at 10:30 Abstract: This talk is a rehearsal for my thesis defense that will take place on Tuesday, May 27th 2025. This thesis focuses on single-phase flow in three-dimensional underground porous media, characterized by fractures, narrow discontinuities that are ubiquitous within the rock matrix. In this work, fractures are specifically considered as preferential flow paths and are modeled using the Discrete Fracture Matrix (DFM) approach. This model preserves the three-dimensional structure of the rock matrix while representing the fracture network as a codimension-one object. The flow is governed by coupled Darcy-type partial differential equations (PDEs), which describe exchanges between the rock and the fractures. While existing literature typically addresses DFMs with a few thousand fractures, this thesis deals with models involving up to several hundreds of thousands. The objective of this work is to design, implement, and analyze a simulation method tailored to such DFMs. The main challenge lies in the efficient solution of the linear systems resulting from the discretization of the PDEs, which may involve several hundreds of millions of degrees of freedom (DOFs). Chapter [1] addresses the mathematical and numerical analysis of the PDE system. It provides the functional framework of the mixed variational formulation and proves the existence and uniqueness of the solution. Mixed finite elements are used for the discretization, for which existence and uniqueness are also demonstrated. A priori error estimates are established as well. The equivalent mixed-hybrid formulation (MHFE) leads to a sparse, square, symmetric, and positive definite linear system. First-order numerical convergence of the MHFE scheme is validated through an academic test case with an analytical solution. Chapter [2] introduces a methodology to assess the performance of linear solvers applied to the system arising from the MHFE discretization presented in Chapter [1]. The fractured networks studied are…

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Geological storage of CO2: an example for the simulation of subsurface flow

Michel Kern: Thursday, 27th March 2025 at 10:30 Abstract: In this talk, I will present the main models used in the simulation of subsurface flow and why they are relevant for the storage of CO2 in deep geological formations. As an example, I will show results from a recent international benchmarking example. The talk is a rehearsal for my “Demi-heure de scicence” presentation on April 3rd.

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Hybrid high-order methods for the wave equation in first-order form

Rekha Khot: Thursday, 20th March 2025 at 10:30 Abstract: In this talk, we will discuss the approximation of the acoustic wave equation in its first-order Friedrichs formulation using hybrid high-order (HHO) methods, proposed and numerically investigated in [Burman-Duran-Ern, 2022]. We first look at energy-error estimates in the time-continuous setting and give several examples of interpolation operators: the classical one in the HHO literature based on L2-orthogonal projections and others from, or inspired from, the hybridizable discontinuous Galerkin (HDG) literature giving improved convergence rates on simplices. The time-discrete setting is based on explicit Runge-Kutta (ERK) schemes in time combined with HHO methods in space. In the fully discrete analysis, the key observation is that it becomes crucial to bound the consistency error in space by means of the stabilization seminorm only. We formulate three abstract properties (A1)-(A3) to lead the analysis. Our main result proves that, under suitable CFL conditions for second- and third-order ERK schemes, the energy error converges optimally in time and quasi-optimally in space, with optimal rates recovered on simplicial meshes. The abstract foundations of our analysis should facilitate its application to other nonconforming hybrid methods such as HDG and weak Galerkin (WG) methods.

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Building (yet other) bridges between polytopal methods

Simon Lemaire: Thursday, 13th March 2025 at 10:30 Abstract: Within the last 20 years or so, a myriad of novel numerical approaches, capable of accommodating general polytopal meshes, have pop up in the literature. The main purpose of this talk is to tidy up the room, and to build connections, in the context of a model variable diffusion problem, between these different approaches. Our study will focus on skeletal methods. As opposed to plain-vanilla finite volume and discontinuous Galerkin discretizations, skeletal methods essentially attach degrees of freedom to the mesh skeleton. Our study will discriminate between primal and mixed formulations of the problem at hand. Somewhat unsurprisingly, we will see that, at the end of the day, all these approaches fall within only two distinct approximation paradigms.

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Shape optimisation using Lipschitz functions

Philip Herbert: Thursday, 06th March 2025 at 10:30 Abstract: In this talk, we discuss a novel method in PDE constrained shape optimisation.  While it is known that many shape optimisation problems have a solution, finding the solution, or an approximation of the solution, may prove non-trivial.  A typical approach to minimisation is to use a first order method; this raises questions when handling shapes – what is a shape derivative, where does it live?  It happens to be convenient to define the derivatives as linear functionals on $W^{1,\infty}$.  We present an analysis of this in a discrete setting along with the existence of directions of steepest descent.  Several numerical experiments will be considered and extensions discussed.

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