13th June – Jean-luc Guermond: Invariant-domain preserving IMEX time stepping methods

Chérif Amrouche Thursday 13th June at 11:00   ABSTRACT: I will present high-order time discretizations of a Cauchy problem where the evolution operator comprises a hyperbolic part and a parabolic part (say diffusion and stiff relaxation terms). The said problem is assumed to possess an invariant domain. I will propose a technique that makes every implicit-explicit (IMEX) time stepping scheme invariant domain preserving and mass conservative. The IMEX scheme is written in incremental form and, at each stage of the scheme, we first compute low-order hyperbolic and parabolic updates, followed by their high-order counterparts. The proposed technique, which is agnostic to the space discretization, allows to optimize the time step restrictions induced by the hyperbolic sub-step. To illustrate the proposed methodology, we derive three novel IMEX schemes with optimal efficiency and for which the implicit scheme is singly-diagonal and L-stable: a third-order, four-stage scheme; and two fourth-order schemes, one with five stages and one with six stages. The novel IMEX schemes are evaluated numerically on a stiff ODE system. We also apply these schemes to nonlinear convection-diffusion problems with stiff reaction and to compressible viscous flows possibly including grey radiation.

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16th June – ‪Chérif Amrouche: Elliptic Problems in Lipschitz and in $C^{1,1}$ Domains

Chérif Amrouche Thursday 16th June at 11:30   ABSTRACT: We are interested here in questions related to the maximal regularity of solutions to elliptic problems with Dirichlet or Neumann boundary conditions (see ([1]). For the last 40 years, many works have been concerned with questions when Ω is a Lipschitz domain. Some of them contain incorrect results that are corrected in the present work. We give here new proofs and some complements for the case of the Laplacian (see [3]), the Bilaplacian ([2] and [6]) and the operator div (A∇) (see ([5]) when A is a matrix or a function. And we extend this study to obtain other regularity results for domains having an adequate regularity. We give also new results for the Dirichlet-to-Neumann operator for Laplacian and Bilaplacian. Using the duality method, we can then revisit the work of Lions-Magenes [4], concerning the so-called very weak solutions, when the data are less regular. References : [1]  C. Amrouche and M. Moussaoui. Laplace equation in smooth or non smooth do- mains. Work in Progress. [2]  B.E.J. Dahlberg, C.E. Kenig, J. Pipher and G.C. Verchota. Area integral estimates for higher-order elliptic equations and systems. Ann. Inst. Fourier, 47-5, 1425– 1461, (1997). [3]  D. Jerison and C.E. Kenig. The Inhomogeneous Dirichlet Problem in Lipschitz Domains, J. Funct. Anal. 130, 161–219, (1995). [4]  J.L. Lions and E. Magenes. Probl`emes aux limites non-homog`enes et applications, Vol. 1, Dunod, Paris, (1969). [5]  J. Necas. Direct methods in the theory of elliptic equations. Springer Monographs in Mathematics. Springer, Heidelberg, (2012). [6]  G.C. Verchota. The biharmonic Neumann problem in Lipschitz domains. Acta Math. 194-2, 217–279, (2005).

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5th May – ‪Daniel Zegarra Vasquez: Simulation of single-phase flows in fractured porous media using the mixed hybrid finite element method

Daniel Zegarra Vasquez Thursday 5th May at 11:00   ABSTRACT: In underground environments, fractures are very numerous and present at all scales, with very heterogeneous sizes. In particular for flows, they are preferential channels: flows are much faster there than in the neighboring rock. Indeed, the permeability of rock is generally about two orders of magnitude lower than that of fractures. This makes fractures play a vital role in a large number of industrial and environmental applications. These particularities of the fractured porous domain make the modeling and simulation of the flows passing through it a major challenge today for which it is necessary to develop dedicated, robust and efficient models and numerical methods. The most commonly used model for representing fractures is the discrete fracture network (DFN) in which fractures are represented as structures of codimension 1. The model of single-phase flows in fractured porous media is described in [5]. The particularity of the fractured porous problem, compared to the porous-only or fractured-only problem [3], is the coupling between the flow in the fractures and the flow in the rock. Due to the difficulties encountered in taking into account the geometric complexity of large fractured networks in simulations, the test cases recently proposed in the literature are mainly 2D, or 3D with a limited number (about ten) of fractures [1]. In this talk, we will present the nef-flow-fpm solver, which solves the stationary 3D fractured porous problem using the mixed hybrid finite element method. The method developed in the solver is inspired by [4]. To mesh the domain, a first simplical and conforming 2D mesh is generated for the DFN and for the boundaries of the domain, then a second simplical and conforming 3D mesh is generated from the first mesh. The solvers integrated in nef-flow-fpm are direct solvers,…

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7th April – ‪Christoph Lehrenfeld: Embedded Trefftz Discontinuous Galerkin methods

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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19th April – ‪Christos Xenophontos: Finite Element approximation of singularly perturbed eigenvalue problems

Christos Xenophontos Tuesday 19th April at 14:00   ABSTRACT: We consider singularly perturbed eigenvalue problems in one-dimension, and their numerical approximation by the (standard Galerkin) Finite Element Method (FEM). These are fourth order equations, where a small parameter multiplies the highest order derivative. We will present results for an $h$ version FEM with polynomials of degree $p$ on an exponentially graded (eXp) mesh, as well as an $hp$ version FEM on the so-called Spectral Boundary Layer (SBL) mesh. For both methods, robust optimal convergence is shown for the eigenvalues and associated eigenfunctions. Numerical results, illustrating the theory, will also be presented. This is joint work with H. G. Roos.

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24th March – ‪Miloslav Vlasak: A posteriori error estimates for discontinuous Galerkin method

Miloslav Vlasak Thursday 24th March at 11:00   ABSTRACT: We will present a posteriori error estimate for higher-order time discretizations, most importantly for the discontinuous Galerkin method, cf. [1]. Rather than the presentation of the estimates themselves, the talk shall focus on the most important ideas behind and their possible application to spatial nonconforming discretizations, most importantly to the discontinuous Galerkin method again. Overall, the talk shall rather focus on open problems than on the presentation of the fully completed results. Additionally, the ideas of reconstructions by the Radau polynomials that are the core ideas in a posteriori error estimates for time discretizations can be exploited for the direct efficiency analysis of the derived estimates. This can enable tracking the dependence of the efficiency constant on the discretization polynomial degree in 1D, cf. [2]. Possible extensions of this result to multiple dimensions shall be discussed. References [1] V. Dolejsi, F. Roskovec, M. Vlasak: A posteriori error estimates for higher order space-time Galerkin discretizations of nonlinear parabolic problems, SIAM J. Numer. Anal. 59, N. 3, 1486–1509 (2021) [2] M. Vlasak: On polynomial robustness of flux reconstructions, Appl. Math. 65, N. 2, 153–172 (2020)

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14th April – ‪Idrissa Niakh: Stable model reduction for linear variational inequalities with parameter-dependent constraints

‪Idrissa Niakh Thursday 14th April at 11:00   ABSTRACT: We consider model reduction for linear variational inequalities with parameter-dependent constraints. We study the stability of the reduced problem in the context of a dualized formulation of the constraints using Lagrange multipliers. Our main result is an algorithm that guarantees inf-sup stability of the reduced problem. The algorithm is computationally effective since it can be performed in the offline phase even for parameter-dependent constraints. Moreover, we also propose a modification of the Cone Projected Greedy algorithm so as to avoid ill-conditioning issues when manipulating the reduced dual basis. Our results are illustrated numerically on the frictionless Hertz contact problem between two half-spheres with parameter-dependent radius and on the membrane obstacle problem with parameter-dependent obstacle geometry.

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10th March – ‪Ruma Maity: Parameter dependent finite element analysis for ferronematics solutions

Ruma Maity‬‬ Thursday 10th March at 11:00   ABSTRACT: In this talk, we focus on the analysis of a free energy functional, that models a dilute suspension of magnetic nanoparticles in a two-dimensional nematic well, referred to as ferronematics. We discuss the asymptotic analysis of global energy minimizers in the limit of vanishing elastic constant, where the re-scaled elastic constant l is inversely proportional to the domain area. The conforming finite element method is used to approximate the regular solutions of the corresponding non-linear system of partial differential equations with cubic nonlinearity and non-homogeneous Dirichlet boundary conditions. We establish the existence and local uniqueness of the discrete solutions, error estimates in the energy and L2 norms with l- discretization parameter dependency. The theoretical results are complemented by the numerical experiments on the discrete solution profiles, and the numerical convergence rates that corroborate the theoretical estimates. This talk is based on joint works with Apala Majumdar (Department of Mathematics and Statistics, University of Strathclyde, UK) and Neela Nataraj (Department of Mathematics, Indian Institute of Technology Bombay, India).

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27th January – ‪Frédéric Lebon‬‬ : On the modeling of nonlinear imperfect solid/solid interfaces by asymptotic techniques

Frédéric Lebon‬‬ Thursday 27th January at 11:00   ABSTRACT: In this talk, we will focus on the theoretical and numerical modelling of interfaces between solids (adhesion, bonding, friction, etc.). We will present a general methodology based on matched asymptotic theory to obtain families of models including the relative rigidity of the interphase (soft or hard), geometrical or material non-linearities (plasticity, curvature, finite strain, …), damage and multiphysics couplings (thermics, piezoelectricity, …).  Numerical results will be presented.

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3rd Feburary – ‪Pierre Matalon‬‬ : An h-multigrid method for Hybrid High-Order discretizations of elliptic equations

Pierre Matalon Thursday 3rd Feburary at 11:00   ABSTRACT: We consider a second order elliptic PDE discretized by the Hybrid High-Order method, for which globally coupled unknowns are located at faces. To efficiently solve the resulting linear system, we propose a geometric multigrid algorithm that keeps the degrees of freedom on the faces at every grid level. The core of the algorithm lies in the design of the prolongation operator that passes information from coarse to fine faces through the reconstruction of an intermediary polynomial of higher degree on the cells. High orders are natively handled by the use of the same polynomial degree at every grid level. The proposed algorithm requires a hierarchy of polyhedral meshes such that the faces (and not only the elements) are successively coarsened. Numerical tests on homogeneous and heterogeneous diffusion problems show fast convergence, asymptotic optimality with respect to the mesh size, robustness to the polynomial order, and robustness with respect to heterogeneity of the diffusion coefficient.

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