Inria, the French national research institute for the digital sciences
Organisation/Company: Inria, the French national research institute for the digital sciences
Research Field: Mathematics
Researcher Profile: First Stage Researcher (R1)
Country: France
Application Deadline: 26 Nov 2024 - 23:00 (UTC)
Type of Contract: Temporary
Job Status: Full-time
Hours Per Week: 38.5
Offer Starting Date: 1 Jan 2025
Is the job funded through the EU Research Framework Programme? Not funded by a EU programme
Reference Number: 2024-08292
Is the Job related to staff position within a Research Infrastructure? No
Offer Description The accurate simulation of time-dependent wave propagation phenomena is central to many areas of physics. Finite element and discontinuous Galerkin methods are very popular for such simulations, since they are able to take complex propagation media into account. The ANR APOWA project aims to improve the reliability and efficiency of these discretization methods by using error estimators and adaptive mesh refinements.
This thesis is part of the APOWA project, funded by the French National Research Agency (ANR). The aim of the thesis is to familiarize the doctoral student with the key themes of the project before embarking on new theoretical developments.
A posteriori error estimation for wave propagation is a delicate subject: [1,3,4] are fundamental contributions that open the way to rigorous numerical analysis. A new approach to a posteriori error estimation of time-dependent wave propagation problems has recently been proposed [2].
The results presented in [2] only take into account the spatial discretization of (1), but for the moment neglect its temporal discretization. The aim of this thesis will be to extend the techniques developed in [2] to take into account the errors associated with temporal discretization.
[1] C. Bernardi, E. Süli, Time and space adaptivity for the second-order wave equation. Math. Models Methods Appl. Sci. 15 (2005), 199-225.
[2] T. Chaumont-Frelet, Asymptotically constant-free and polynomial-degree-robust a posteriori estimates for space discretizations of the wave equation. SIAM J. Sci. Comput. 45 (2023), A1591-A1620.
[3] T. Chaumont-Frelet, A. Ern, M. Vohrali´k, On the derivation of guaranteed and p-robust a posteriori error estimates for the Helmholtz equation, Numer. Math. 148 (2021), 525-573.
[4] W. Dörfler and S.A. Sauter, A posteriori error estimation for highly indefinite Helmholtz problems, Comput. Methods Appl. Math. 13 (2013), 333-347.
This thesis project has three objectives:
The PhD student will develop an a posteriori error estimator that takes into account the spatiotemporal discretization of the wave equation. This estimator will be implemented in a one-dimensional toy calculation code.
He or she will develop a two-dimensional implementation to evaluate the performance of the error estimator for complex configurations. He or she will empirically use the estimator in adaptive mesh refinement algorithms.
Finally, he or she will analyze from a theoretical point of view the adaptive computation algorithms previously implemented from a practical point of view.
We are looking for a candidate with a Master's degree or an engineering diploma in applied mathematics. He or she should have knowledge of PDE theory, numerical analysis and the finite element method. We are also particularly interested in candidates who have already worked on wave propagation and/or a posteriori error estimation, as well as those with programming skills.
Languages: FRENCH Level Basic
Languages: ENGLISH Level Good
Additional Information Public transport partially reimbursed
Vacations: 7 weeks' annual leave + 10 days' RTT (full-time basis) + possibility of exceptional leave (e.g. sick children, moving house)
Possibility of telecommuting and flexible working hours
Professional equipment available (videoconferencing, loan of computer equipment, etc.)
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