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Computer-assisted proofs of radial solutions of elliptic systems on R^d

Olivier Hénot： Thursday 23rd Nov at 11:00 The talk presents recent work on the rigorous computation of localized radial solutions of semilinear elliptic systems. While there are comprehensive results for scalar equations and some specific classes of elliptic systems, much less is known about these solutions in generic systems of nonlinear elliptic equations. These radial solutions are described by systems of non-autonomous ordinary differential equations. Using an appropriate Lyapunov-Perron operator, we rigorously enclose the centre-stable manifold, which contains the asymptotic behaviour of the profile. We then formulate, as a zero-finding problem, a shooting scheme from the set of initial conditions onto the invariant manifold. By means of a Newton-Kantorovich-type theorem, we obtain sufficient conditions to prove the existence and local uniqueness of a zero in the vicinity of a numerical approximation. We apply this method to prove ground state solutions for the Klein-Gordon equation on R^3, the Swift-Hohenberg equation on R^2, and a FitzHugh-Nagumo system on R^2.

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A Volume-Preserving Reference Map Method for the Level Set Representation

Maxime Theillard： Thursday 16th Nov at 17:00 Abstract: This seminar will present an implicit interface representation, where the geometry is captured by a level set function, and its deformations are reconstructed from the diffeomorphism between the warped and original geometries (the reference map). A key advantage of this representation is that it provides a local estimation of numerical local mass losses. Using this metric, we design a novel projection for the reference map on the space of volume- preserving diffeomorphisms, which results in enhanced but inexact, mass conservation. In the limit of small deviations from this space, the projection is shown to be uniquely defined, and the correction can be computed as the solution of a Poisson problem. The method is analyzed and validated in two and three spatial dimensions. Both the theoretical and computational results show it excels at correcting the mass loss due to inaccuracy in the advection process or the velocity field. This error reduction is particularly impactful for practical applications, such as the simulation of multiphase flows over long time intervals, and offers improved computational exploration capabilities.

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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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Implementing $H^2$-conforming finite elements without enforcing $C^1$-continuity

Zhaonan DongInternal Seminar 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.

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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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