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

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Simulation for the Environment: Reliable and Efficient Numerical Algorithms

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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.

Charles Parker: Monday 13th Nov at 11:00am ABSTRACT: Fourth-order elliptic problems arise in a variety of applications from thin plates to phase separation to liquid crystals. A conforming Galerkin discretization requires a finite dimensional subspace of $H^2$, which in turn means that conforming finite element subspaces are $C^1$-continuous. In contrast to standard $H^1$-conforming $C^0$ elements, $C^1$ elements, particularly those of high order, are less understood from a theoretical perspective and are not implemented in many existing finite element codes. In this talk, we address the implementation of the elements. In particular, we present algorithms that compute $C^1$ finite element approximations to fourth-order elliptic problems and which only require elements with at most $C^0$-continuity. We show that the resulting subproblems are uniformly stable with respect to the mesh size and polynomial degree in 2D and illustrate the method on a number of representative test problems.

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.

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.

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).

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. .

Simon Legrand: Thursday 2nd February at 11:00 ABSTRACT: While essential in most scientific fields, parameter studies can be tedious and error-prone if they are not led with proper tools. Prune_rs (Prune in Rust) is a tool/language aimed at easily describing complex parameter spaces and automatizing the execution of commands over each parameter combination. It also offers the user predefined patterns to store results and simplifies postprocessing. Prune_rs is an ongoing development process and we would be glad to discuss your suggestions to make it better!

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.

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.