## About

80

Publications

11,579

Reads

**How we measure 'reads'**

A 'read' is counted each time someone views a publication summary (such as the title, abstract, and list of authors), clicks on a figure, or views or downloads the full-text. Learn more

1,557

Citations

Citations since 2016

Introduction

My interest in Applied Mathematics lies in devising and analyzing numerical methods and mathematical models for multiscale phenomena appearing in engineering and life science problems. My favorite tools in mathematics are numerical analysis, asymptotic analysis, partial differential equations, finite element methods and domain decomposition techniques. I am also interested in the implementation of numerical algorithms underlying multiscale numerical methods for massive parallel architectures.

Additional affiliations

May 2015 - May 2017

September 2013 - January 2015

**INRIA Sophia Antipolis**

Position

- Visiting Researcher

June 2005 - July 2006

**University of Colorado at Denver**

Education

March 1995 - December 1998

**Universite Pierre et Marie Curie - Paris VI**

Field of study

- Applied Mathematics

March 1992 - July 1994

**Universidade Federal do Rio de Janeiro - UFRJ**

Field of study

- Mathematics

March 1988 - December 1991

## Publications

Publications (80)

The multiscale hybrid-mixed (MHM) method consists of a multi-level strategy to approximate the solution of boundary value problems with heterogeneous coefficients. In this context, we propose a new family of finite elements for the linear elasticity equation defined on coarse polytopal partitions of the domain. The finite elements rely on face degr...

Physics-Informed Neural Networks (PINNs) are machine learning tools that approximate the solution of general partial differential equations (PDEs) by adding them in some form as terms of the loss/cost function of a Neural Network. Most pieces of work in the area of PINNs tackle non-linear PDEs. Nevertheless, many interesting problems involving line...

We establish the equivalence between the Multiscale Hybrid-Mixed (MHM) and the Multiscale Hybrid High-Order (MsHHO) methods for a variable diffusion problem with piecewise polynomial source term. Under the idealized assumption that the local problems defining the multiscale basis functions are exactly solved, we prove that the equivalence holds for...

We establish the equivalence between the Multiscale Hybrid-Mixed (MHM) and the Multiscale Hybrid High-Order (MsHHO) methods for a variable diffusion problem with piecewise polynomial source terms. Under the idealized assumption that the local problems defining the multiscale basis functions are exactly solved, we prove that the equivalence holds fo...

This work proposes a family of multiscale hybrid-mixed methods for the two-dimensional linear elasticity problem on general polygonal meshes. The new methods approximate displacement, stress, and rotation using two-scale discretizations. The first scale level setting consists of approximating the traction variable (Lagrange multiplier) in discontin...

A novel residual a posteriori error estimator for the Oseen equations achieves efficiency and reliability by including multilevel contributions in its construction. Originates from the Multiscale Hybrid Mixed (MHM) method, the estimator combines residuals from the skeleton of the first-level partition of the domain, along with the contributions fro...

The multiscale hybrid-mixed (MHM) method consists of a multi-level strategy to approximate the solution of boundary value problems with heterogeneous coefficients. In this context, we propose a new family of finite elements for the linear elasticity equation defined on coarse polytopal partitions of the domain. The finite elements rely on face degr...

This work extends the general form of the multiscale hybrid-mixed (MHM) method for the second-order Laplace (Darcy) equation to general non-conforming polygonal meshes. The main properties of the MHM method, i.e., stability, optimal convergence, and local conservation, are proven independently of the geometry of the elements used for the first leve...

Multiscale Hybrid Mixed (MHM) method refers to a numerical technique targeted to approximate systems of differential equations with strongly varying solutions. For fluid flow, normal fluxes (multiplier) over macro element boundaries, and coarse piecewise constant potential approximations in each macro element are computed (upscaling). Then, small d...

This work proposes a novel multiscale finite element method for acoustic wave propagation in highly heterogeneous media which is accurate on coarse meshes. It originates from the primal hybridization of the Helmholtz equation at the continuous level, which relaxes the continuity of the unknown on the skeleton of a partition. As a result, face-based...

This work extends the general form of the Multiscale Hybrid-Mixed (MHM) method for the second-order Laplace (Darcy) equation to general non-conforming polygonal meshes. The main properties of the MHM method, i.e., stability, optimal convergence, and local conservation, are proven independently of the geometry of the elements used for the first leve...

This work proposes and analyzes a residual a posteriori error estimator for the Multiscale Hybrid-Mixed (MHM) method for the Stokes and Brinkman equations. The error estimator relies on the multi-level structure of the MHM method and considers two levels of approximation of the method. As a result, the error estimator accounts for a first-level glo...

Multiscaled Hybrid Mixed (MHM) method refers to a numerical technique targeted to approximate systems of differential equations with strongly varying solutions. For fluid flows, normal fluxes (multiplier) over macro element boundaries, and coarse piecewise constant potential approximations in each macro element are computed (upscaling). Then, small...

In this work, we address time-dependent wave propagation problems with strong multiscale features (in space and time). Our goal is to design a family of innovative high-performance numerical methods suitable for the simulation of such multiscale problems. Particularly, we extend the Multiscale Hybrid-Mixed finite element method (MHM for short) for...

The Multiscale Hybrid Mixed methods (MHM for short) are a family of multiscale methods that naturally incorporate multiple scales in the numerical solution, while providing high-order approximations for both primal and dual variables. The multiscale feature is a consequence of a hybridization procedure, that replaces the former problem in a set of...

The family of Multiscale Hybrid-Mixed (MHM) finite element methods has received considerable attention from the mathematics and engineering community in the last few years. The MHM methods allow solving highly heterogeneous problems on coarse meshes while providing solutions with high-order precision. It embeds independent local problems which are...

In this work, we address time dependent wave propagation problems with strong multiscale features (in space and time). Our goal is to design a family of innovative high performance numerical methods suitable to the simulation of such multiscale problems. Particularly, we extend the Multiscale Hybrid-Mixed finite element method (MHM for short) for t...

An abstract setting for the construction and analysis of the Multiscale Hybrid-Mixed (MHM for short) method is proposed. We review some of the most recent developments from this standpoint, and establish relationships with the classical lowest-order Raviart-Thomas element and the primal hybrid method, as well as with some recent multiscale methods....

The multiscale hybrid-mixed (MHM) method is extended to the Stokes and Brinkman equations with highly heterogeneous coefficients. The approach is constructive. We first propose an equivalent dual-hybrid formulation of the original problem using a coarse partition of the heterogeneous domain. Faces may be not aligned with jumps in the data. Then, th...

In this work we prove uniform convergence of the Multiscale Hybrid-Mixed (MHM for short) finite element method for second order elliptic problems with rough periodic coefficients. The MHM method is shown to avoid resonance errors without adopting oversampling techniques. In particular, we establish that the discretization error for the primal varia...

A new family of finite element methods, named Multiscale Hybrid-Mixed method (or MHM for short), aims to solve reactive-advective dominated problems with multiscale coefficients on coarse meshes. The underlying upscaling procedure transfers to the basis functions the responsibility of achieving high orders of accuracy. The upscaling is built inside...

This work proposes and analyzes a new local projection stabilized (LPS for short) finite element method for the nonlinear incompressible Navier-Stokes equations. Stokes problems defined element-wisely drive the construction of the stabilized terms which make the present method stable for P1 × P1, for continuous pressure and P1 × P0 for discontinuou...

This work presents a family of stable finite element methods for two-and three-dimensional linear elasticity models. The weak form posed on the skeleton of the partition is a byproduct of the primal hybridization of the elasticity problem. The unknowns are the piecewise rigid body modes and the Lagrange multipliers used to relax the continuity of d...

This work proposes and analyses an adaptive finite element scheme for the fully non-linear incompressible Navier-Stokes equations. A residual a posteriori error estimator is shown to be effective and reliable. The error estimator relies on a Residual Local Projection (RELP) finite element method for which we prove well-posedness under mild conditio...

This work presents a priori and a posteriori error analyses of a new multiscale hybrid-mixed method (MHM) for an elliptic model. Specially designed to incorporate multiple scales into the construction of basis functions, this finite element method relaxes the continuity of the primal variable through the action of Lagrange multipliers, while assuri...

We restrict the variational multiscale method to a class of methods we denote by numerical multiscale methods. Numerical multiscale methods are methods obtained by enriching the piecewise linear functions with special local functions. The enrichment provides additional stabilization via terms obtained by static condensation. The resulting methods a...

This work combines two complementary strategies for solving the steady incompressible Navier–Stokes model with a zeroth-order term, namely, a stabilized finite element method and a mesh–refinement approach based on an error estimator. First, equal order interpolation spaces are adopted to approximate both the velocity and the pressure while stabili...

A Galerkin enriched finite element method (GEM) is proposed for the singularly perturbed reaction-diffusion equation. This new method is an improvement on the Petrov-Galerkin enriched method (PGEM), where now the standard piecewise (bi)linear test space incorporates fine scales. This appears as the fundamental ingredient for suppressing oscillation...

A Galerkin enriched finite element method (GEM) is proposed for the singularly perturbed reaction–diffusion equation. This new method is an improvement on the Petrov–Galerkin enriched method (PGEM), where now the standard piecewise (bi)linear test space incorporates fine scales. This appears as the fundamental ingredient for suppressing oscillation...

This work presents and analyzes a new residual local projection stabilized finite element method (RELP) for the nonlinear incompressible Navier–Stokes equations. Stokes prob-lems defined elementwise drive the construction of the residual-based terms which make the present method stable for the finite element pairs P 1 /P l , l = 0, 1. Numerical upw...

The development of new numerical methods is of great importance in computational science. Due to their many appealing properties, Finite Element (FEMs), Finite Volume (FVMs) and Finite Difference (DFMs) methods are of particular interest, with a very large number of journal articles devoted to them. Unfortunately, these methods can be time consumin...

We derive two stabilized methods for transient equations using static condensation of residual-free bubbles. The methods enhance the stability of the Discontinuous Galerkin method.

This work proposes a new local projection stabilized finite element method (LPS) for the Oseen problem. The method adds to the Galerkin formulation new fluctuation terms that are symmetric and easily computable at the element level. Proposed for the pair ℙ1/ℙl, l = 0, 1, when the pressure is continuously or discontinuously approximated, well-posedn...

The simplest pair of spaces ₁/P₂ is made inf-sup stable for the mixed form of the Darcy equation. The key ingredient is to enhance the finite element spaces inside a Petrov-Galerkin framework with functions satisfying element-wise local Darcy problems with right hand sides depending on the residuals over elements and edges. The enriched method is s...

This paper presents the development of a database to support ecological data management and treatment. This database, developed in the context of the PELD project, is provided with inference capability, achieved from the relational model mapping to RDF. Taxonomical, spatial and trophic relations are explored by means of nested sets and explicit tra...

A new residual local projection stabilized method (RELP) is proposed as a result of an enriched Petrov-Galerkin strategy for the Oseen problem. The P 2 1 × P l pairs, l = 0, 1 with continuous or discontinuous pressures, are made stable by enhancing them with solutions of residual-based local Oseen problems and performing a static condensation proce...

This work establishes a formal derivation of local projection stabilized methods as a result of an enriched Petrov-Galerkin strategy for the Stokes problem. Both velocity and pressure finite element spaces are enhanced with solutions of residual- based local problems, and then the static condensation procedure is applied to derive a new stable and...

The aim of this paper is twofold. First, we review the recent Petrov–Galerkin enriched method (PGEM) to stabilize numerical solutions of BVP's in primal and mixed forms. Then, we extend such enrichment technique to a mixed singularly perturbed problem, namely, the generalized Stokes problem, and focus on a stabilized finite element method arising i...

A new symmetric local projection method built on residual bases (RELP) makes linear equal-order finite element pairs stable for the Darcy problem. The derivation is performed inside a Petrov–Galerkin enriching space approach (PGEM) which indicates parameter-free terms to be added to the Galerkin method without compromising consistency. Velocity and...

We discuss the numerical integration of polynomials times non-polynomial weight- ing functions in two dimensions arising from multiscale finite element computa- tions. The proposed quadrature rules are significantly more accurate than standard quadratures and are better suited to existing finite element codes than formulas computed by symbolic inte...

This work introduces and analyzes novel stable Petrov-Galerkin Enriched Meth- ods (PGEM) for the Darcy problem based on the simplest but unstable continuous P1/P0 pair. Stability is recovered inside a Petrov-Galerkin framework where element-wise depen- dent residual functions, named multi-scale functions, enrich both velocity and pressure trial spa...

Starting from the non-stable P1/P0 discretization we build enhanced methods for the Darcy equation which are stable and locally mass-conservative. The methods are derived in a Petrov–Galerkin framework where both velocity and pressure trial spaces are enriched with multiscale functions. These functions solve local problems correcting the residuals...

A new stabilized finite-element method is presented for the Stokes problem. The method is of a Douglas–Wang type, and includes
a positive jump term controlling the residual of the Cauchy stress tensor on the internal edges of the triangulation. A priori
error estimates are obtained in the natural norms of the unknowns and an a posteriori error esti...

Eective boundary conditions (wall laws) are commonly employed to approximate PDEs in domains with rough boundaries, but it is neither easy to design such laws nor to estimate the related approximation error. A two-scale asymptotic expansion based on a domain decomposition result is used here to mitigate such diculties, and as an application we cons...

This work concerns the development of stabilized finite element methods for the Stokes problem considering nonstable different (or equal) order of velocity and pressure interpola- tions. The approach is based on the enrichment of the standard polynomial space for the velocity component with multiscale functions which no longer vanish on the element...

This work concerns the derivation of new stabilized finite element methods for the Stokes problem. Starting from pairs of spaces which are not stable, they are made stable by enriching them with multiscale functions, i.e., functions which are local, but not bubble-like, arising from the solution of local problems at the element level. This general...

We derive a new stabilized finite element method for the generalized Stokes problem starting from the non-stable continuous P1/P1 finite element space enriched with multiscale functions. The stabilization parameter is related with the enrichment func- tions which are analytically computed from a boundary value problem at the element level leading t...

In this paper we propose a novel way, via finite elements to treat problems that can be singular perturbed, a reaction–diffusion equation in our case. We enrich the usual piecewise linear or bilinear finite element trial spaces with local solutions of the original problem, as in the residual free bubble (RFB) setting, but do not require these funct...

We develop enriched finite element methods for parabolic singularly perturbed problems. A Petrov–Galerkin strategy is employed, where the test and the trial spaces are enriched with ‘bubble’ and multiscale functions, respectively. We examine two semi-discrete formulations for the unsteady reaction–diffusion equation, namely the method of lines and...

We introduce a hierarchic a posteriori error estimate for singularly perturbed reaction–diffusion problems. The estimator is based on a Petrov–Galerkin method in which the trial space is enriched with nonpolynomial functions or multiscale functions. We study the equivalence between the a posteriori estimate and the exact error in the energy norm. M...

We perform an error analysis for a multiscale finite element method for singularly perturbed reaction-diffusion equations. Such a method is based on enriching the usual piecewise linear finite element trial spaces with local solutions of the original problem, but the approach does not require these functions to vanish on each element edge. Piecewis...

We propose a finite element method based on enriching the Galerkin approximation spaces with a combination of multiscale functions and residual-free bubbles (RFB). This approach is presented as a Petrov-Galerkin method and applied to the singularly perturbed reaction-advection-diffusion model. Numerical tests confirm that switching RFB by suitable...

We propose a multiscale finite element method to treat singularly perturbed reaction diffusion equations. We enrich the usual piecewise linear or bilinear finite el-ement trial spaces with local solutions of the original problem, as in the Residual Free Bubble (RFB) setting, but do not require these functions to vanish on each element edge. Such mu...

An unusual stabilized finite element is presented and analyzed herein for a generalized Stokes problem with a dominating zeroth order term. The method consists in subtracting a mesh dependent term from the formulation without compromising consistency. The design of this mesh dependent term, as well as the stabilization parameter involved, are sugge...

The aim of this work is the construction of effective boundary conditions (wall laws) for elliptic problems defined in domains with curved, rough boundaries with periodic wrinkles. We present error estimates for first and second order approximations, and a numerical test. To cite this article: A. Madureira, F. Valentin, C. R. Acad. Sci. Paris, Ser....

Different effective boundary conditions or wall laws for unsteady incompressible Navier-Stokes equations over rough domains are derived in the laminar setting. First and second order unsteady wall laws are proposed using two scale asymptotic expansion techniques. The roughness elements are supposed to be periodic and the influence of the rough boun...

An improved unusual finite element method is studied herein for a second-order linear scalar differential equation including a zero order term. The method consists in subtracting from the standard Galerkin method a mesh dependent term suggested by static condensation of the bubbles. Based on this idea, a new stabilized parameter is constructed, whi...

We propose effective boundary conditions or wall laws for a laminar flow over a rough wall with periodic roughness elements. These effective conditions are posed on a regularized boundary which allows to avoid the details of the wall and dramatically reduces computational cost. The effective conditions stem from an asymptotic expansion of the solut...

This paper is concerned with the numerical simulation of flows of viscous fluids over rough surfaces. The proposed methodology introduces consistant equivalent wall laws using Domain Decomposition techniques with nonmatching grids and local periodicity assumptions. The paper describes this approach with its theoretical background, studies its limit...

We describe a new approach for developing new wall-laws for rough
surfaces. We also give error estimates on a simple model.

We combine the Galerkin-least-squares (GLS) and the Galerkin-gradient-leastsquares (GGLS) methods to simulate scalar linear second order partial differential equations which include second, first and zero order terms. Assuming a strictly positive coefficient for the second order term, the resulting method is proven convergent for a wide range of th...