Internal Seminar

Roland Maier: Thursday 9th February at 11:00 ABSTRACT: This talk is about semi-explicit time-stepping schemes for coupled elliptic-parabolic problems as they arise, for instance, in the context of poroelasticity. If the coupling between the equations is rather weak, such schemes may be applied. Their main advantage is that they decouple the equations and therefore allow for faster computations. Theoretical convergence results are presented that rely on a close connection of the semi-explicit schemes to partial differential equations that include delay terms. Numerical experiments confirm the theoretical findings.

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Fabio Vicini Thursday 17th Nov at 11:00   ABSTRACT: The presentation will focus on some numerical aspects of the simulations of porous-fractured media I have experienced so far. Flow simulations on porous fractured media contain many numerical challenging issues related to complex geometries and the severe size of the computational domain. In real applications, indeed, the network of fractures immersed in the rock matrix is usually generated according to known probabilistic distributions, which lead to a huge number of fractures with a multi-scale distribution in sizes. For these reasons, tailored numerical methods are s to overcome these problems and for efficient handling of the computational resources. I will show two different approaches to tackle the classic Darcy single-phase problem with the Discrete Fracture Networks (DFNs) model: one related to a constrained optimization problem and the second related to the modern Virtual Element (VEM) framework. Moreover, I will present the possibility of investigating the Darcy problem on DFNs as a parameterized differential problem with a Reduced Basis (RB) technique combined with a-posteriori error control. Finally, I will discuss the possibility of extending the VEM approach for the simulation of a two-phase flow of immiscible fluids.

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Xuefeng Liu Monday 28th Nov at 11:00   ABSTRACT: We are concerned with the guaranteed computation of the eigenfunction (or eigenspace) of differential operators. Upon the setting of eigenvalue problems, three algorithms are proposed to give rigorous and efficient error estimation for the approximate eigenfunctions: Algorithm I: Rayleigh quotient error based algorithm; Algorithm II: Variational residual error based algorithm; Algorithm III: Projection error based algorithm. The guaranteed eigenfunction computation is applied to solving shape optimization problems. For example, the minimization of the Laplacian eigenvalue over polygonal domains. By explicitly evaluating the Hadamard shape derivative with guaranteed computation of both eigenvalues and eigenfunctions, we provide a computer-assisted proof to declare that the equilateral triangle minimizes the first Laplacian under certain non-homogeneous Neumann boundary condition under the radius constraint condition. References: 1. R. Endo and X. Liu, Shape optimization for the Laplacian eigenvalue over triangles and its application to interpolation error constant estimation, https://arxiv.org/abs/2209.13415. 2. X. Liu and T. Vejchodsky, Projection error-based guaranteed L2 error bounds for finite element approximations of Laplace eigenfunctions, https://arxiv.org/abs/2211.03218. 3.X. Liu and T. Vejchodsky, Fully computable a posteriori error bounds for eigenfunctions, Numer. Math. 152 (2022), 183–221.

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Iuliu Sorin Pop Thursday 20th Oct at 10:00   ABSTRACT: We discuss various aspects related to non-equilibrium mathematical models for porous media flow. Typically, such models assume that quantities like saturation, phase pressure differences, or relative permeability are related by monotone, algebraic relationships. Under such assumptions, the solutions satisfy the maximum principle. On the other hand, experimental work published in the past decades report that phenomena like saturation overshoot, or the formation of finger profiles have been observed whenever the flow is sufficiently rapid. Such results are ruled out by standard, equilibrium models. This is the main motivation to consider non-equilibrium models, where dynamic or hysteretic effects are included in the above-mentioned relationships. The resulting models are nonlinear evolution systems of (pseudo-)parabolic and possibly degenerate equations, and involving differential inclusions. For such problems, we present first some results concerning the derivation of such models from the pore scale to the Darcy scale, the existence and uniqueness of weak solutions, and discuss different numerical schemes. This includes aspects like the rigorous convergence of the discretization, and solving the emerging nonlinear time-discrete or fully discrete problems. This is joint work with X. Cao (Toronto), S. Karpinski (Munich), S. Lunowa (Munich), K. Mitra (Hasselt), F.A. Radu (Bergen)

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Rekha Khot Thursday 06th Oct at 15:00   ABSTRACT: The lowest-order nonconforming virtual element extends the Morley triangular element to polygons for the approximation of the weak solution $u \in  V := H_0^2(Ω)$ to the biharmonic equation. Two nonconforming virtual element spaces have been introduced in [1, 4] for $H^3$ regular solutions, while a medius analysis in [3] allows minimal regularity. In this talk, we will discuss an abstract framework (cf. [2]) with two hypotheses (H1)-(H2) for unified stability and a priori error analysis of at least two different discrete spaces (even a mixture of those). A smoother J allows rough source terms $F \in V^∗ = H^{−2}(\Omega)$. The a priori and a posteriori error analysis circumvents any trace of second derivatives by a computable conforming companion operator $J : V_h \rightarrow V$ from the nonconforming virtual element space $V_h$. The operator J is a right-inverse of the interpolation operator and leads to optimal error estimates in piecewise Sobolev norms without any additional regularity assumptions on $$u \in V$$. As a smoother the companion operator modifies the discrete right-hand side and then allows a quasi-best approximation. An explicit residual-based a posteriori error estimator is reliable and efficient up to data oscillations. Numerical examples display the predicted empirical convergence rates for uniform and optimal convergence rates for adaptive mesh-refinement. [1] P. F. Antonietti, G. Manzini, and M. Verani, The fully nonconforming virtual element method for biharmonic problems, Math. Models Methods Appl. Sci., 28 (2018), pp. 387–407. [2] C. Carstensen, R. Khot, and A. K. Pani, Nonconforming virtual elements for the biharmonic equation with Morley degrees of freedom on polygonal meshes, arXiv:2205.08764, (2022). [3] J. Huang and Y. Yu, A medius error analysis for nonconforming virtual element methods for Poisson and biharmonic equations, J. Comput. Appl. Math., 386 (2021), p. 21. [4] J. Zhao, B. Zhang,…

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Alexandre IMPERIALE Wednesday 21th Sep at 15:00   ABSTRACT: The Department of Imaging and Simulation for Control (DISC) at CEA – LIST is dedicated to developing Non-Destructive Testing (NDT) methods in advanced industrial contexts (e.g. nuclear energy, aeronautic, or petrochemical industries). Amongst different testing modalities, Ultrasonic Testing (UT) is widespread and plays a major role in numerous industrial processes. It relies on the propagation of mechanical waves in order to probe critical specimens, often formed with complex media – in terms of geometry or material properties. To fully understand the capabilities and performances of UT experiments, the DISC is developing modelling tools for wave propagation, which are bound to incorporate the CIVA platform [1] – a commercial software dedicated to NDT modelling. When commercializing modelling tools, efficiency and robustness are key features. This talk focuses on numerical methods for wave propagation problems that address these challenges. In particular, we show how the combination of Spectral Finite Elements [2] with the Mortar Element method [3] – a domain decomposition approach – can be tuned to render efficient and robust algorithms [4]. Coupled with high-frequency asymptotic methods [5] (e.g. ray-based algorithms), this combination forms the basis of various advances in the CIVA platform. This approach is typically used to modelling UT experiments relying on bulk wave propagation, e.g. for testing welds (typical nuclear energy industry applications) or composite materials [6] (typical aeronautic industry applications). Additionally, we present some details on ongoing works related to other UT configurations, such as numerical modelling of wave propagation through thin layers – based on effective transmission conditions – or within plate-like geometries – achieved through dedicated time schemes. [1] http://www.extende.com/civa-in-a-few-words [2] Cohen, G. C. “Higher-order numerical methods for transient wave equations”. Berlin: Springer, 2002 Vol. 5. [3] F. Ben Belgacem, Y. Maday, “The mortar element method…

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Daniel Zegarra Vasquez Thursday 5th May at 11:00   ABSTRACT: Dans les milieux souterrains, les fractures sont très nombreuses et présentes à toutes les échelles, avec des tailles très hétérogènes. Notamment pour les écoulements, elles sont des voies préférentielles : les écoulements y sont beaucoup plus rapides que dans la roche avoisinante. En effet, la perméabilité de la roche est généralement environ deux ordres de grandeur plus faible que celle des fractures. Cela fait que les fractures jouent un rôle primordial dans un grand nombre d’applications industrielles et environnementales. Ces particularités du domaine poreux fracturé rendent la modélisation et la simulation des écoulements qui y transitent un défi majeur aujourd’hui pour lequel il convient de développer des modélisations et méthodes numériques dédiées robustes et efficaces. Le modèle le plus couramment utilisé de représentation des fractures est le modèle de réseaux de fractures discrets dans lequel les fractures sont représentées comme des structures de codimension 1. Le modèle d’écoulements monophasiques en milieux poreux fracturés est décrit dans [5]. La particularité du problème poreux fracturé, par rapport au problème uniquement poreux ou uniquement fracturé [3], est le couplage entre l’écoulement dans les fractures et l’écoulement dans la roche. Du fait des difficultés rencontrées pour prendre en compte la complexité géométrique de grands réseaux fracturés dans les simulations, les cas tests proposés récemment dans la littérature sont majoritairement 2D, ou 3D avec un nombre limité (une dizaine) de fractures [1]. Dans cet exposé, nous présenterons le solveur nef-flow-fpm, qui permet de résoudre le problème poreux fracturé 3D stationnaire grâce à la méthode des éléments finis mixtes hybrides. La méthode développée dans le solveur est inspirée de [4]. Pour mailler le domaine, un premier maillage 2D simplical et conforme est généré pour le réseau de fractures et pour les bords du domaine, puis un…

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Christoph Lehrenfeld Thursday 7th April at 11:00   ABSTRACT: Discontinuous Galerkin (DG) methods are widely used to discretize partial differential equations (PDEs) due to (a.o.) flexibility for designing robust methods and simplicity in terms of data structures. One major drawback of DG methods is, however, the increased number of (globally coupled) degrees of freedom (ndof) compared to, for instance, continuous Galerkin methods. One – by now established – remedy is the use of Hybrid DG methods. These allow reducing the globally coupled ndof essentially by introducing the concept of static condensation for DG methods. Thereby the dimensions of global linear systems that need to be solved for reduced from O(p^d) to O(p^{d−1}), where d is the space dimension and p is the polynomial degree of the finite element space. A different approach is the use of Trefftz DG methods, where a DG formulation is modified by restricting the finite element spaces to functions that element-wise solve the PDE at hand. This results in a similar reduction of globally coupled ndof, cf. Figure 1 below. However, due to several limitations, Trefftz DG methods have only been applied for special PDEs so far. On the one hand, the finite element spaces have to be specifically tailored for each PDE type, on the other hand, inhomogeneous equations and non-constant coefficients in the differential operators are difficult to deal with and rarely treated. In this talk, we introduce Embedded Trefftz DG methods which exploit the existence of an underlying standard DG formulation for an efficient and flexible implementation of Trefftz DG methods. Furthermore, we relax Trefftz DG methods in view of the constraints on the finite element space leading to weak Trefftz DG spaces. Both together allow us to remove the limitations of Trefftz DG methods in order to enable them for a large…

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