Category: Internal Seminar

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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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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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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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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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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