Vincent Fontaine

• PhD
• Professor (Assistant) at University of Reunion Island

Discontinuous Galerkin (DG) methods due to their robustness properties, e.g. local conservation, low numerical dispersion, and well-capturing strong shocks and physical discontinuities, are well-suited for the simulation of Variable Density Flow (VDF) in porous media. This paper aims at introducing, in a unified format, the general class of Interio...

We analyze families of primal high-order hybridizable discontinuous Galerkin (HDG) methods for solving degenerate (second-order) elliptic problems. One major trouble regarding this class of PDEs concerns its mathematical nature, which may be nonuniform over the domain. Due to the local degeneracy of the diffusion term, it can be purely hyperbolic i...

In this paper, we present a hybridizable discontinuous Galerkin (HDG) mixed method for second-order diffusion problems using a projective stabilization function and broken Raviart–Thomas functions to approximate the dual variable. The proposed HDG mixed method is inspired by the primal HDG scheme with reduced stabilization suggested by Lehrenfeld a...

The present paper focuses on the numerical modeling of groundwater flows in fractured porous media using the codimensional model description. Therefore, fractures are defined explicitly as a (d − 1)-dimensional geometric object immersed in a d-dimensional region and can act arbitrarily as a drain or a barrier. We numerically investigate a novel num...

We analyze families of primal high-order hybridized discontinuous Galerkin (HDG) methods for solving degenerate second-order elliptic problems. One major problem regarding this class of PDEs concerns its mathematical nature, which may be nonuniform over the whole domain. Due to the local degeneracy of the diffusion term, it can be purely hyperbolic...

During the last decades, modeling flows in fractured porous media has received tremendous attention from engineering, geophysical, and research fields. We distinguish the micro-fractures that are covered by homogenization or upscaling techniques, and large fractures that act as preferential paths or barriers to the flow. In the latter case, the fra...

The present paper discusses families of Interior Penalty Discontinuous Galerkin (IP) methods for solving heterogeneous and anisotropic diffusion problems. Specifically, we focus on distinctive schemes, namely the Hybridized-, Embedded-, and Weighted-IP schemes, leading to final matrixes of different sizes and sparsities. Both the Hybridized- and Em...

Modeling fluid flow in fractured porous media has received tremendous attention from engineering, geophysical, and other research fields over the past decades. We focus here on large fractures described individually in the porous medium, which act as preferential paths or barriers to the flow.

In this paper, we derive improved a priori error estimates for families of hybridizable interior penalty discontinuous Galerkin (H-IP) methods using a variable penalty for second-order elliptic problems. The strategy is to use a penalization function of the form $\mathcal{O}(1/h^{1+\delta})$, where $h$ denotes the mesh size and $\delta$ is a user-d...

This poster describe a numeral method to solve diffusion-convection problems in strong heterogeneous/anisotropic medium

This paper deals with global sensitivity analysis of computer model output. Given an independent input sample and associated model output vector with possibly the vector of output derivatives with respect to the input variables, we show that it is possible to evaluate the following global sensitivity measures: (i) the Sobol’ indices, (ii) the Borgo...

In this work, lowest-order Raviart–Thomas and Brezzi–Douglas–Marini mixed methods are considered for groundwater flow simulations. Typically, mixed methods lead to a saddle-point problem, which is expensive to solve. Two approaches are numerically compared here to allow an explicit velocity elimination: (1) the well-known hybrid formulation leading...

Mixed finite element (MFE) and multipoint flux approximation (MPFA) methods have similar properties and are well suited for the resolution of Darcy's flow on anisotropic and heterogeneous domains. In this work, the link between hybrid and MPFA formulations is shown algebraically for the lowest order mixed methods of Raviart–Thomas (RT0) and Brezzi–...

The mixed hybrid finite element (MHFE) and the multipoint flux approximation (MPFA) methods are well suited for anisotropic heterogeneous domains since both are locally conservative and can handle general irregular grids. In this work, behaviours and performances of MHFE and MPFA methods are studied numerically for different heterogeneities and ani...