Recent preprint

• PC and N. Heuer. A DPG framework for strongly monotone operators (2018), [Hal]
  
  Short abstract A new hybrid technique to approximate the very-weak solution of a nonlinear strongly monotone problem is presented and analyzed. This scheme is based on the use of optimal test functions and nonlinear penalty method.

Published articles

• PC. Well-posedness of the scalar and the vector advection-reaction problems in Banach graph spaces, [Hal]
  C. R. Math. Acad. Sci. Paris, 355(8), 892--902 (2017)
  Short abstract An extension of the well-posedness of advection-reaction problems is proposed in Banach graph spaces and under more general assumptions on the physical parameters is proposed. In particular we consider the analysis for divergence-free advection fields.
  
  
• PC and A. Ern. An edge-based scheme on polyhedral meshes for vector advection-reaction equations, [Hal]
  ESAIM Math. Mod. Numer. Anal., 51(5), 1561--1581 (2017)
  Short abstract A new scheme using scalar-valued edge dofs is presented and analyzed to approximate the vector advection-reaction problem. The discrete analysis is done under general assumptions on the parameters, including e.g. divergence-free fields. Numerical results are presented on 3D polyhedral meshes.
  
  
• PC, J. Bonelle, E. Burman and A. Ern. A vertex-based scheme on polyhedral meshes for advection-reaction equations with sub-mesh stabilization, [Hal]
  Comput. Math. Appl., 72(9), 2057--2071 (2016)
  Short abstract This work present the first vertex-based scheme on polyhedral meshes approximating the advection-reaction problem at order 3/2. An extension of the lowest Lagrange finite element on polyhedral meshes is introduced. Various test cases from the literature are considered on 3D polyhedral meshes.
  
  
• PC and A. Ern. Vertex-Based Compatible Discrete Operator Schemes on Polyhedral Meshes for Advection-Diffusion Equations, [Hal]
  Comput. Meth. in Appl. Math., 16(2), 187--212 (2016)
  Short abstract In this work, we present a CDO scheme for the scalar advection-diffusion problem. This scheme preserves the structure of the problem and the behavior of the solution when the Peclet number goes from zero to infinity. Numerical experiemnts are presented on 3D polyhedral meshes.

Articles in preparation

• PC and N. Heuer. A penalized Petrov-Galerkin method for pseudo-monotone problems
• PC and N. Heuer. Robust dPG method for a nonlinear convection-dominated elliptic problem

Some recent talks

• A nonlinear least-square framework for strongly monotone operators
• Well-posedness of advection-reaction problems in Banach spaces
• An edge-based scheme on polyhedral meshes for the vector advection-reaction problem
• A compact stencil scheme on polyhedral meshes for steady transport problems

Posters

• Peclet robust CDO schemes for advection-diffusion problems on polyhedral meshes
• A new accurate scheme on polyhedral meshes for transport problems