[1] J. Dabaghi, and G. Delay, A unified framework for high-order numerical discretizations of variational inequalities. Computers & Mathematics with Applications (2021). Journal version , preprint .

[2] J. Dabaghi, V. Martin, and M. Vohralik, Adaptive inexact semismooth Newton methods for the contact problem between two membranes. J. Sci. Comput. (2020). DOI 10.1007/s10915-020-01264-3. Journal version , preprint .

[3] J. Dabaghi, V. Martin, and M. Vohralik, A posteriori estimates distinguishing the error components and adaptive stopping criteria for numerical approximations of parabolic variational inequalities. Comput. Methods Appl. Mech. Engrg. 367 (2020), 113105. Journal version , preprint .

[4] I. Ben Gharbia, J. Dabaghi, V. Martin, and M. Vohralik, A posteriori error estimates and adaptive stopping criteria for a compositional two-phase flow with nonlinear complementarity constraints. Comput. Geosci. 24 (2020), 1031-1055. DOI 10.1007/s10596-019-09909-5 . Journal version , preprint .

[5] J. Dabaghi, V. Ehrlacher, and C. Strossner, Computation of the self-diffusion coefficient with low-rank tensor methods: application to the simulation of a cross-diffusion system. Esaim: Proceedings and Surveys, (2022). preprint .

Papers submitted

Papers in preparation

• A code to compute the numerical solutions of cross-diffusion systems by finite volumes and a code constructing a POD-reduced order model for cross-diffusion systems zip code .

Some results on cross-diffusion systems




In the first three images is represented the evolution of the concentrations of 3 species as a function of time. The concentrations evolve to constant profiles. In the last image is plotted the behavior of two reduced order model error. A standard POD error and a structure preserving SP error.
     



Some results on a parareal-in-time resolution for speeding a Monte Carlo resolution dedicated to the heat equation.


In the following images is represented a parareal resolution for the heat equation. In the parareal resolution, the coarse propagator is the finite element solver and the fine propagator is the Monte Carlo solver. Few parareal iterations are needed to speed up the resolution.
 



Some results for high order discretizations of variational inequalities


In the following images, is represented the error in the energy norm for a smooth solution (not realistic case) and compare the finite element method and the HHO method.
   


In the following images, is represented the error in the energy norm for a realistic solution (discontinuous Lagrange multiplier) and compare the finite element method and the HHO method.
   

In the following images, is represented the concept of adaptivity for inexact semismooth Newton resolutions of variational inequalities. The first image shows the overall behavior of the estimators as a function of the number of mesh elements. For exact and inexact semismooth Newton resolutions the discretization estimator is dominant and the linearization and algebraic estimators are small. In the adaptive inexact semismooth Newton resolution, the linearization and algebraic estimators are sufficiently small to not influence the behavior of the discretization estimator. The second image focuses on the behavior of the different error estimators as a function of the linear algebra (GMRES) resolution. An important number of algebraic iterations can be saved. The third image shows the overall performance of three methods.
   

• IMAG: invited by Prof Daniele Di Pietro , University of Montpellier, November 2021, France.


• CEMRACS, July-August 2021, Luminy, France.
  Research project: Computation of the auto-diffusion coefficient for a cross diffusion problem with tensor methods.
  In collaboration with Virginie Ehrlacher and Christoph Strossner

Collaborations