PhD Position F/M Mesh and solvers adaptivity for nonlinear partial differential equations: contraction and optimality

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Type de contrat : CDD

Niveau de diplôme exigé : Bac + 5 ou équivalent

Fonction : Doctorant

Contexte et atouts du poste

French–German ANR–DFG project RANPDEs. France: joint research team SERENA https://team.inria.fr/serena/, Inria Paris, and Ecole nationale des ponts et chaussées. Germany: Institute for Numerical Simulation, University of Bonn, https://ins.uni-bonn.de/.

Mission confiée

The proposed Ph.D. position concerns numerical approximation of nonlinear partial differential equations. We will consider a model second-order partial differential equation with strongly monotone and Lipschitz continuous nonlinearity, stemming from a nonlinear convex energy minimization problem. The ultimate target is to compute a discrete (piecewise polynomial) approximation of the unknown exact solution with error below the given desired tolerance at the expense of minimal computational cost.

A crucial aspect is the encompassment of both nonlinear solvers (such as Zarantonello, Picard, or Newton) and linear solvers (such multigrid) in addition to the finite element discretization (a practical necessity seldom addressed theoretically). This reflects the specificity of the subject at the interplay between analysis of partial differential equations, numerical analysis, and numerical linear algebra. The algorithms are designed relying on a posteriori estimates of the computational error.

Principales activités

The goals are

  • Prove contraction of the energy difference in each mesh adaptation step.
  • Establish optimal error decay rates with respect to the number of unknowns (no algorithm can achieve a better convergence rate wrt the number of unknowns).
  • Establish optimal error decay rates with respect to the computational cost (no algorithm can achieve a better convergence rate wrt the computational cost, defined as a cumulated sum of the number of unknowns appearing on each algorithm step).
  • Implement the resulting algorithm on a computer nd assess it numerically on model problems with known solution.

Compétences

Master in mathematics, ideally with focus on numerical analysis and scientific computing (finite element or finite volume methods, iterative linearization methods, iterative algebraic solvers).

Programming skills (C, C++, Matlab, Python or Julia).

Avantages

  • Subsidized meals
  • Partial reimbursement of public transport costs
  • Leave: 7 weeks of annual leave + 10 extra days off due to RTT (statutory reduction in working hours) + possibility of exceptional leave (sick children, moving home, etc.)
  • Possibility of teleworking (after 6 months of employment) and flexible organization of working hours
  • Professional equipment available (videoconferencing, loan of computer equipment, etc.)
  • Social, cultural and sports events and activities
  • Access to vocational training
  • Social security coverage