Computer-assisted proofs for nonlinear equations: how to turn a numerical simulation into a theorem.

Maxime Breden: Thursday 9th Nov at 11:00am Abstract: The goal of a posteriori validation methods is to get a quantitative and rigorous description of some specific solutions of nonlinear dynamical sys- tems, often ODEs or PDEs, based on numerical simulations. The general strategy consists in combining a priori and a posteriori error estimates, in- terval arithmetic, and a fixed point theorem applied to a quasi-Newton op- erator. Starting from a numerically computed approximate solution, one can then prove the existence of a true solution in a small and explicit neigh- borhood of the numerical approximation. I will first present the main ideas behind these techniques on a simple example, and then describe the results of a recent joint work with Jan Bouwe van den Berg and Ray Sheombarsing, in which we use these techniques to rigorously enclose solutions of some parabolic PDEs.

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A posteriori error estimates robust with respect to nonlinearities and final time.

Martin Vohralík: Thursday, 25th May at 11:00 ABSTRACT: A posteriori estimates enable us to certify the error committed in a numerical simulation. In particular, the equilibrated flux reconstruction technique yields a guaranteed error upper bound, where the flux obtained by local postprocessing is of independent interest since it is always locally conservative. In this talk, we tailor this methodology to model nonlinear and time-dependent problems to obtain estimates that are robust, i.e., of quality independent of the strength of the nonlinearities and the final time. These estimates include and build on common iterative linearization schemes such as Zarantonello, Picard, Newton, or M- and L-ones. We first consider steady problems and conceive two settings: we either augment the energy difference by the discretization error of the current linearization step, or we design iteration-dependent norms that feature weights given by the current iterate. We then turn to unsteady problems. Here we first consider the linear heat equation and finally move to the Richards one, which is doubly nonlinear and exhibits both parabolic–hyperbolic and parabolic–elliptic degeneracies. Robustness with respect to the final time and local efficiency in both time and space are addressed here. Numerical experiments illustrate the theoretical findings all along the presentation.

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On the preconditioned Newton’s method for Richards’ equation

Konstantin Brenner: Thursday, 11th May at 11:00 ABSTRACT: Richards’ equation is arguably the most popular hydrogeological flow model, which can be used to predict the underground water movement under both saturated and unsaturated conditions. However, despite its importance for hydrogeological applications, this equation is infamous for being difficult to solve numerically. Indeed, depending on the flow parameters, the resolution of the systems arising after the discretization may become an extremely challenging task, as the linearization schemes such as Picard or Newton’s methods may fail or exhibit unacceptably slow convergence. In this presentation, I will first give a brief overview of Richards’ equation both from the hydrogeological and mathematical perspectives. Then we will discuss the nonlinear preconditioning strategies that can be used to improve the performance of Newton’s method. In this regard, I will present some traditional techniques involving the primary variables substitution as well as some recent ones based on the nonlinear Jacobi or block Jacobi preconditioning. The later family of (block) Jacobi-Newton methods turn out to be a very attractive option as they allow for the global convergence analysis in the framework of the Monotone Newton Theorem.

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High order exponential fitting discretizations for convection diffusion problems

Ludmil Zikatanov: Thursday, 4th May at 11:00 ABSTRACT: We discuss discretizations for convection diffusion equations in arbitrary spatial dimensions. Targeted applications include the Nernst-Plank equations for transport of species in a charged media. We illustrate how such exponentially fitted methods are derived in any spatial dimension. A main step in proving error estimates is showing unisolvence for the quasi-polynomial spaces of differential forms defined as weighted spaces of differential forms with polynomial coefficients. We show that the unisolvent set of functionals for such spaces on a simplex in any spatial dimension is the same as the set of such functionals used for the polynomial spaces. We are able to prove our results without the use of Stokes’ Theorem, which is the standard tool in showing the unisolvence of functionals in polynomial spaces of differential forms. This is joint work with Shuonan Wu (Beijing University).

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23rd March – Marien-Lorenzo Hanot: Polytopal discretization of advanced differential complexes.

Marien-Lorenzo Hanot: Thursday, 23rd March at 11:00 ABSTRACT: We are interested in the discretization of advanced differential complexes. That is to say, complexes presenting higher regularity or additional algebraic constraints compared to the De Rham complex.This type of complex appears naturally in the discretization of many systems of differential equations. For example, the Stokes complex uses the same operators as the De Rham complex. Still, it requires an increased regularity, or the Div-Div complex appears in biharmonic equations and requires the use of fields with values in symmetric or traceless matrices. The principle of polytopal methods is to use discrete functions not belonging to a subset of the continuous functions but are composed of a collection of polynomials defined on objects of any dimension of the mesh (on edges, faces, cells…).This allows using very generic meshes, in our case composed of arbitrary contractible polytopes, while keeping the computability of discrete functions. The objective is to present the construction of a family of discrete 3-dimensional Div-Div complexes for arbitrary polynomial degrees. These complexes are consistent on polynomial functions, which is the basis for obtaining an optimal convergence of the schemes built on them. Moreover, they preserve the algebraic structure of the continuous complex, in the sense that the cohomology of the discrete complex is isomorphic to that of the continuous. .

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

Jean-Luc Guermond 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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20 January – ‪Isabelle Ramiere‬ : Automatic multigrid adaptive mesh refinement with controlled accuracy for quasi-static nonlinear solid mechanics

Isabelle Ramiere‬ Thursday 20th January at 11:00   ABSTRACT: Many real industrial problems involve localized effects (nonlinearity, contact, heterogenity,…). Adaptive Mesh Refinement (AMR) approaches are well-suited numerical techniques to take into account mesoscale phenomena in simulation processes. For implicit solvers (such as for quasi-static mechanics problems), classical h and/or p-adaptive refinement strategies consisting in generating a unique global mesh locally refined (in mesh step and/or in degree of basis function) are limited by the resulting size of problems to be solved (cf. number of DoFs). Hence, we were interested in local multigrid methods, consisting in adding local refined nested meshes in zones of interest without modifying the initial computation mesh. An iterative process (similar to standard multigrid solvers) enables to correct to various levels solutions. We have extended the multigrid Local Defect Correction (LDC) method (Hackbusch, 1984), initially introduced in Computational Fluid Dynamics, to elastostaticity (Barbié et al., 2014) with a multilevel generalization of the algorithm. In order to automatically detect the zone of interest and hence to avoid the pollution error, the LDC method has been coupled with an a posteriori error estimate of Zienckiewicz-Zhu type (Barbié et al., 2014; Barbié et al., 2015; Liu et al., 2017). We also proposed an original stopping criterion in case of local singularity (Ramière et al., 2019). We have compared in (Koliesnikova et al.,2021) within a unified AMR framework the efficiency of the LDC method with respect to conforming and nonconforming h-adaptive strategies. We have also extended the LDC method to structural mechanics nonlinearities. In (Liu et al., 2017), an efficient algorithm has been developed in order to deal with frictional contact via the LDC method. For nonlinear material behaviours, a one time step algorithm has been first introduced in (Barbié et al., 2015) while a fully automatic algorithm in time with…

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25 November – Pierre Gosselet: Asynchronous Global/Local coupling

Pierre Gosselet: Thursday 25th November at 11:00   ABSTRACT: Non-intrusive global/local coupling can be seen as an exact iterative version of the submodeling (structural zoom) technique widely used by industry in their simulations. A global model, coarse but capable of identifying general trends in the structure, is locally patched by fine models with refined geometries, materials and meshes. The coupling is achieved by alternating Dirichlet resolutions on the patches and global resolutions with a well-chosen immersed Neumann condition. After the preliminary work of (Whitcomb, 1991), the method has been rediscovered by many authors. Our work starts with (Gendre et al., 2009). From a theoretical point of view, the method is related to the optimized Schwarz domain decomposition methods (Gosselet et al., 2018). It has been applied in many contexts (localized or generalized (visco)plasticity, stochastic calculations, cracking, damage, fatigue…). In the ANR project ADOM, we are working on the implementation of an asynchronous version of the method. The expected benefits of asynchronism (Magoulès et al., 2018; Glusa et al., 2020) are to reach the solution faster, to adapt to many computational hardware by being more resilient in case of poor load balancing, network latencies or even outages. During the presentation, I will show how to adapt the global/local coupling to asynchronism and will illustrate its performance on thermal and linear elasticity calculations. This work is realized with the support of National Research Agency, project [ANR-18-CE46-0008]. [1] Gendre, Lionel et al. (2009). “Non-intrusive and exact global/local techniques for structural problems with local plasticity”. In: Computational Mechanics 44.2, pp. 233–245. [2] Glusa, Christian et al. (2020). “Scalable Asynchronous Domain Decomposition Solvers”. In: SIAM Journal on Scientific Computing 42.6, pp. C384–C409. doi: 10.1137/19M1291303. [3] Gosselet, Pierre et al. (2018). “Non-invasive global-local coupling as a Schwarz domain decomposition method: acceleration and generalization”. In: Advanced Modeling and…

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24 November – Grégory Etangsale: A primal hybridizable discontinuous Galerkin method for modelling flows in fractured porous media

Grégory Etangsale: Wednesday 24th November at 10:30   ABSTRACT: Modeling fluid flow in fractured porous media has received tremendous attention from engineering, geophysical, and other research fields over the past decades. We focus here on large fractures described individually in the porous medium, which act as preferential paths or barriers to the flow. Two different approaches are available from a computational aspect: The first one, and definitively the oldest, consists of meshing inside the fracture. In this case, the flow is governed by a single Darcy equation characterized by a large scale of variation of the permeability coefficient within the matrix region and the fracture, respectively. However, this description becomes quite challenging since it requires a considerable amount of memory storage, severely increasing the CPU time. A more recent approach differs by considering the fracture as an encapsulated object of lower dimension, i.e., (d − 1)-dimension. As a result, the flow process is now governed by distinctive equations in the matrix region and fractures, respectively. Thus, coupling conditions are added to close the problem. This mathematical description of the fractured porous media has been initially introduced by Martin et al. in [4] and is referred to as the Discrete Fracture-Matrix (DFM) model. The DFM description is particularly attractive since it significantly simplifies the meshing of fractures and allows the coupling of distinctive discretizations such as Discontinuous and Continuous Galerkin methods inside the bulk region and the fracture network, respectively. For instance, we refer the reader to the recent works of Antonietti et al. [1] (and references therein), where the authors coupled the Interior Penalty DG method with the (standard) H1-Conforming finite element method to solve the DFM problem (see e.g., [3]). However, it is well-known that DG methods are generally more expensive than most other numerical methods due to their high…

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