Raphael Bulle

National Institute for Research in Computer Science and Control | INRIA · MIMESIS

This manuscript is concerned with a posteriori error estimation for the finite element discretization of standard and fractional partial differential equations as well as an application of fractional calculus to the modeling of the human meniscus by poro-elasticity equations. In the introduction, we give an overview of the literature of a posterior...

We develop a novel a posteriori error estimator for the L2 error committed by the finite element discretization of the solution of the fractional Laplacian. Our a posteriori error estimator takes advantage of the semi-discretization scheme using a rational approximation which allows to reformulate the fractional problem into a family of non-fractio...

In this study, we observe that the poromechanical parameters in human meniscus vary spatially throughout the tissue. The response is anisotropic and the porosity is functionally graded. To draw these conclusions, we measured the anisotropic permeability and the “aggregate modulus” of the tissue, i.e., the stiffness of the material at equilibrium, a...

In the seminal paper of Bank and Weiser [Math. Comp., 44 (1985), pp.283-301] a new a posteriori estimator was introduced. This estimator requires the solution of a local Neumann problem on every cell of the finite element mesh. Despite the promise of Bank-Weiser type estimators, namely locality, computational efficiency, and asymptotic sharpness, t...

In this poster, we present our new hierarchical a posteriori error estimation method for the spectral fractional Laplacian equation with homogeneous Dirichlet boundary condition. The numerical results show the adaptively refined mesh we obtain using our method as well as a convergence comparison where we can notice the sharpness of the estimator an...

We provide a new argument proving the reliability of the Bank-Weiser estimator for Lagrange piecewise linear finite elements in both dimension two and three. The extension to dimension three constitutes the main novelty of our study. In addition, we present a numerical comparison of the Bank-Weiser and residual estimators for a three-dimensional te...

We provide a new argument proving the reliability of the Bank-Weiser estimator for Lagrange piecewise linear finite elements in both dimension two and three. The extension to dimension three constitutes the main novelty of our study. In addition, we present a numerical comparison of the Bank-Weiser and residual estimators for a three-dimensional te...

This work has been initiated during the Maths-Companies study group (Semaine d’Étude Maths-Entreprise, SEME) organised by AMIES (Agence pour les Mathematiques en Interaction avec l’Entreprise et la Société), CNRS (Centre National de la Recherche Scientifique) and the University of Strasbourg from 12th to 18th of November 2018. The subject was propo...