Juan Galvis

Juan Galvis
National University of Colombia | UNAL · Departamento de Matemáticas (Bogotá)

PhD

About

87
Publications
7,067
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
2,520
Citations
Introduction
I am an applied mathematician working mainly in analysis and numerical analysis of partial differential equations related to fluid flows in porous media. Other interests include applications of numerical methods for the solution of interesting problems in different areas.
Additional affiliations
July 2013 - July 2013
King Abdullah University of Science and Technology
Position
  • Business Visitor
August 2011 - July 2012
Institute for Scientific Computation (ISC)
Position
  • PostDoc Position
August 2008 - May 2011
Texas A&M University
Position
  • Professor (Assistant)
Education
March 2004 - February 2008
March 2002 - February 2004

Publications

Publications (87)
Article
Full-text available
An abstract framework for constructing stable decompositions of the spaces corresponding to general symmetric positive definite problems into "local" subspaces and a global "coarse" space is developed. Particular applications of this abstract framework include practically important problems in porous media applications such as: the scalar elliptic...
Article
In this paper we study multiscale finite element methods (MsFEMs) using spectral multiscale basis functions that are designed for high-contrast problems. Multiscale basis functions are constructed using eigenvectors of a carefully selected local spectral problem. This local spectral problem strongly depends on the choice of initial partition of uni...
Article
Full-text available
In this paper we design an iterative domain decomposition method for free boundary problems with nonlinear flux jump condition. Our approach is related to damped Newton's methods. The proposed scheme requires, in each iteration, the approximation of the flux on (both sides of) the free interface. We present a Finite Element implementation of our me...
Article
Full-text available
In this paper, we present a high-order expansion for elliptic equations in high-contrast media. The background conductivity is taken to be one and we assume the medium contains high (or low) conductivity inclusions. We derive an asymptotic expansion with respect to the contrast and provide a procedure to compute the terms in the expansion. The comp...
Preprint
Full-text available
In this paper, we present numerical procedures to compute solutions of partial differential equations posed on fractals. In particular, we consider the strong form of the equation using standard graph Laplacian matrices and also weak forms of the equation derived using standard length or area measure on a discrete approximation of the fractal set....
Preprint
Full-text available
In this note, we analyze a procedure used in predictive security applications that motivates the definition of the integral average transform. Our motivation comes from the need for a better mathematical understanding of the prediction accuracy index. This index is used to identify hot spots in predictive security and other applications. We present...
Preprint
Full-text available
We consider parametric families of partial differential equations--PDEs where the parameter $\kappa$ modifies only the (1,1) block of a saddle point matrix product of a discretization below. The main goal is to develop an algorithm that removes, as much as possible, the dependence of iterative solvers on the parameter $\kappa$. The algorithm we pro...
Article
Full-text available
The goal of this work is to present a fast and viable approach for the numerical solution of the high-contrast state problems arising in topology optimization. The optimization process is iterative, and the gradients are obtained by an adjoint analysis, which requires the numerical solution of large high-contrast linear elastic problems with featur...
Article
Using oblique projections and angles between subspaces we write condition number estimates for abstract nonsymmetric domain decomposition methods. In particular, we consider a restricted additive method for the Poisson equation and write a bound for the condition number of the preconditioned operator. We also obtain the non-negativity of the precon...
Preprint
Full-text available
A regularized fracture model for porous media is proposed. Our model is obtained through homogenization theory and formal asymptotic expansions applied to a regularized brittle fracture model considering a perforated material.
Preprint
Full-text available
The goal of this work is to present a fast and viable approach for the numerical solution of the high-contrast state problems arising in topology optimization. The optimization process is iterative, and the gradients are obtained by an adjoint analysis, which requires the numerical solution of large high-contrast linear elastic problems with featur...
Preprint
Full-text available
We present and discuss a novel approach to deal with conservation properties for the simulation of nonlinear complex porous media flows in the presence of: 1) multiscale heterogeneity structures appearing in the elliptic-pressure-velocity and in the rock geology model, and 2) multiscale wave structures resulting from shock waves and rarefaction int...
Article
In this paper, we address the numerical homogenization approximation of a free-boundary dam problem posed in a heterogeneous media. More precisely, we propose a Generalized Multiscale Finite Element Method (GMsFEM) for the heterogeneous dam problem. The motivation of using the GMsFEM approach comes from the multiscale nature of the porous media due...
Article
In this paper, we consider an approximation method, and a novel general analysis, for second-order elliptic differential equations with heterogeneous multiscale coefficients. We obtain convergence of the Generalized Multi-scale Finite Element Method (GMsFEM) method that uses local eigenvectors in its construction. The analysis presented here can be...
Preprint
Full-text available
Using oblique projections and angles between subspaces we write condition number estimates for abstract nonsymmetric domain decomposition methods. In particular, we design and estimate the condition number of restricted additive Schwarz methods. We also obtain non-negativity of the pre-conditioner operator. Condition number estimates are not enough...
Conference Paper
Cattle movement represents one of the principal risks for foot-and-mouth disease (FMD) propagation. The characterization of this complex transportation network may aid in surveillance and control tasks. In particular, network centrality may provide relevant information for FMD epidemiology. Several centrality measures can be computed for the cattle...
Preprint
Full-text available
In this paper, we address the numerical homogenization approximation of a free-boundary dam problem posed in a heterogeneous media. More precisely, we propose a generalized multiscale finite element (GMsFEM) method for the heterogeneous dam problem. The motivation of using the GMsFEM approach comes from the multiscale nature of the porous media due...
Preprint
In this paper, we consider an approximation method, and a novel general analysis, for second-order elliptic differential equations with heterogeneous multiscale coefficients. We obtain convergence of the Generalized Multi-scale Finite Element Method (GMsFEM) method that uses local eigenvectors in its construction. The analysis presented here can be...
Article
Full-text available
We solve elliptic systems of equations posed on highly heterogeneous materials. Examples of this class of problems are composite structures and geological processes. We focus on a model problem which is a second-order elliptic equation with discontinuous coefficients. These coefficients represent the conductivity of a composite material. We assume...
Article
We solve elliptic systems of equations posed on highly heterogeneous materials. Examples of this class of problems are composite structures and geological processes. We focus on a model problem which is a second-order elliptic equation with discontinuous coefficients. These coefficients represent the conductivity of a composite material. We assume...
Article
Full-text available
In this paper we put together some tools from differential topology and analysis in order to study second oder semi-linear partial differential equations on a Riemannian manifold $M$. We seek for solutions that are constants along orbits of a given group action. Using some results obtained by Helgason in [J DIFFER GEOM,6(3), 411-419] we are able to...
Chapter
In this paper, we explore a method for the construction of locally conservative flux fields. The flux values are obtained through the use of a Ritz formulation in which we augment the resulting linear system of the continuous Galerkin (CG) formulation in a higher-order approximation space. These methodologies have been successfully applied to multi...
Chapter
We review some important ideas in the design and analysis of robust overlapping domain decomposition algorithms for high-contrast multiscale problems.
Article
Full-text available
We review some important ideas in the design and analysis of robust overlapping domain decomposition algorithms for high-contrast multiscale problems and propose a domain decomposition method better performance in terms of the number of iterations. The main novelty of our approaches is the construction of coarse spaces, which are computed using spe...
Article
Full-text available
We use a Monte Carlo method to assemble finite element matrices for polynomial Chaos approximations of elliptic equations with random coefficients. In this approach, all required expectations are approximated by a Monte Carlo method. The resulting methodology requires dealing with sparse block-diagonal matrices instead of block-full matrices. This...
Article
Full-text available
A new high-order conservative finite element method for Darcy flow is presented. The key ingredient in the formulation is a volumetric, residual-based, based on Lagrange multipliers in order to impose conservation of mass that does not involve any mesh dependent parameters. We obtain a method with high-order convergence properties with locally cons...
Article
In this manuscript apply some recent results on the study of problems with high-contrast coefficients to linear elasticity problem. We derive an asymptotic expansion for solutions of heterogeneous elasticity problems in terms of the high-contrast in the coefficients. We also study the convergence of the expansion in the $H^1$ norm.
Article
Full-text available
We study the one-dimensional Riemann problem for a hyperbolic system of three conservation laws of Temple class. This systems it is a simplification of a recently propose system of five conservations laws by Bouchut and Boyaval that model viscoelastic fluids. An important issues is that the considered $3 \times 3$ system is such that every characte...
Chapter
In this paper we consider a boundary value problem for elliptic second order partial differential equations with highly discontinuous coefficients in a 2D polygonal region Ω. The problem is discretized by a (full) DG method on triangular elements using the space of piecewise linear functions. The goal of this paper is to study a special version of...
Article
Full-text available
Abstract. In this article we prove the existence of at least one positive solution for a three-point integral boundary-value problem for a second-order nonlinear differential equation. The existence result is obtained by using Schauder’s fixed point theorem. Therefore, we do not need local assumptions such as superlinearity or sublinearity of the i...
Article
Full-text available
Abstract. In this article we prove the existence of at least one positive solution for a three-point integral boundary-value problem for a second-order nonlinear differential equation. The existence result is obtained by using Schauder’s fixed point theorem. Therefore, we do not need local assumptions such as superlinearity or sublinearity of the i...
Article
In this paper, we propose a method for the construction of locally conservative flux fields through a variation of the Generalized Multiscale Finite Element Method (GMsFEM). The flux values are obtained through the use of a Ritz formulation in which we augment the resulting linear system of the continuous Galerkin (CG) formulation in the higher-ord...
Data
We study linear elasticity problems with high contrast in the coefficients using asymptotic limits recently introduced. We derive an asymptotic expansion to solve heterogeneous elasticity problems in terms of the contrast in the coefficients. We study the convergence of the expansion in the H1 norm.
Article
In this paper, we present and analyze a FETI-DP solver with deluxe scaling for a Nitsche-type discretization [Comput. Methods Appl. Math. 3 (2003), 76–85], [SIAM J. Numer. Anal. 49 (2011), 1761–1787] based on a discontinuous Galerkin (DG) method for elliptic two-dimensional problems with discontinuous coefficients and non-matching meshes only acros...
Article
In this paper, we propose a multiscale empirical interpolation method for solving nonlinear multiscale partial differential equations. The proposed method combines empirical interpolation techniques and local multiscale methods, such as the Generalized Multiscale Finite Element Method (GMsFEM). To solve nonlinear equations, the GMsFEM is used to re...
Article
Full-text available
In this paper, we study the development of efficient multiscale methods for flows in heterogeneous media. Our approach uses the Generalized Multiscale Finite Element (GMsFEM) framework. The main idea of GMsFEM is to approximate the solution space locally using a few multiscale basis functions. This is typically achieved by selecting an appropriate...
Article
In this paper, we combine discrete empirical interpolation techniques, global mode decomposition methods, and local multiscale methods, such as the Generalized Multiscale Finite Element Method (GMsFEM), to reduce the computational complexity associated with nonlinear flows in highly-heterogeneous porous media. To solve the nonlinear governing equat...
Article
Full-text available
In this paper, we propose a multiscale empirical interpolation method for solving nonlinear multiscale partial differential equations. The proposed method combines empirical interpolation techniques and local multiscale methods, such as the Generalized Multiscale Finite Element Method (GMsFEM). To solve nonlinear equations, the GMsFEM is used to re...
Conference Paper
We present a Boundary Element Method (BEM)-based FEM for mixed formulations of second order elliptic problems in two dimensions. The challenge, we would like to address, is a proper construction of \(\mathbf H(\mathrm {div})\)–conforming vector valued trial functions on arbitrary polygonal partitions of the domain. The proposed construction generat...
Article
Full-text available
We study the one-dimensional Riemann problem for a hyperbolic system of three conservation laws of Temple class. This system is a simplification of a recently proposed system of five conservations laws by Bouchut and Boyaval that model viscoelastic fluids. An important issue is that the considered 3 × 3 system is such that every characteristic fi...
Article
In this paper we consider the numerical upscaling of the Brinkman equation in the presence of high-contrast permeability fields. We develop and analyze a robust and efficient Generalized Multiscale Finite Element Method (GMsFEM) for the Brinkman model. In the fine grid, we use mixed finite element method with the velocity and pressure being continu...
Article
In this paper, we study multiscale finite element methods for Richards' equation, a mathematical model to describe fluid flow in unsaturated and highly heterogeneous porous media. In order to compute solutions of Richard's equation, one can use numerical homogenization or multiscale methods that use two-grid procedures: a fine-grid that resolves th...
Article
In this paper, multiscale finite element methods (MsFEMs) and domain decomposition techniques are developed for a class of nonlinear elliptic problems with high-contrast coefficients. In the process, existing work on linear problems [Y. Efendiev, J. Galvis, R. Lazarov, S. Margenov and J. Ren, Robust two-level domain decomposition preconditioners fo...
Article
Full-text available
In this paper a discretization based on discontinuous Galerkin (DG) method for an elliptic two-dimensional problem with discontinuous coefficients is considered. The problem is posed on a polygonal region $\Omega$ which is a union of $N$ disjoint polygonal subdomains $\Omega_i$ of diameter $O(H_i)$. The discontinuities of the coefficients, possibly...
Article
We consider multiscale flow in porous media. We assume that we can characterize the ensemble of all possible flow scenarios, that is, we can describe all possible permeability configurations needed for the simulations. We construct coarse basis functions that can provide inexpensive coarse approximations that are: (1) adequate for all possible flow...
Article
Full-text available
In this paper, we present the Multiscale Finite Element Method (MsFEM) for problems on rough heterogeneous surfaces. We consider the diffusion equation on oscillatory surfaces. Our objective is to represent small-scale features of the solution via multiscale basis functions described on a coarse grid. This problem arises in many applications where...
Article
Full-text available
In this paper we use the GeneralizedMultiscale Finite ElementMethod (GMsFEM) framework, introduced in [20], in order to solve nonlinear elliptic equations with high-contrast coefficients. The proposed solution method involves linearizing the equation so that coarse-grid quantities of previous solution iterates can be regarded as auxiliary parameter...
Article
Full-text available
In this paper, we propose oversampling strategies in the Generalized Multiscale Finite Element Method (GMsFEM) framework. The GMsFEM, which has been recently introduced in [12], allows solving multiscale parameter-dependent problems at a reduced computational cost by constructing a reduced-order representation of the solution on a coarse grid. The...
Article
Motivated by applications to numerical simulation of flows in highly heterogeneous porous media, we develop multiscale finite element methods for second order elliptic equations. We discuss a multiscale model reduction technique in the framework of the discontinuous Galerkin finite element method. We propose three different finite element spaces on...
Article
In this paper, we propose a general approach called Generalized Multiscale Finite Element Method (GMsFEM) for performing multiscale simulations for problems without scale separation over a complex input space. As in multiscale finite element methods (MsFEMs), the main idea of the proposed approach is to construct a small dimensional local solution...
Article
In this paper a Nitsche-type discretization based on a discontinuous Galerkin (DG) method for an elliptic two-dimensional problem with discontinuous coefficients is considered. The problem is posed on a polygonal region Ω which is a union of N disjoint polygonal subdomains Ωi of diameter O(Hi). The discontinuities of the coefficients, possibly very...
Article
Full-text available
In this paper, we study robust two-level domain decomposition preconditioners for highly anisotropic multiscale problems. We present a construction of coarse spaces that emploies initial multiscale basis functions and discuss techniques to achieve smaller dimensional coarse spaces without sacrificing the robustness of the preconditioner. We also pr...
Article
Full-text available
Motivated by applications to numerical simulations of flows in highly heterogeneous porous media, we develop multiscale finite element methods for second order elliptic equations. We discuss a multiscale model reduction technique in the framework of the discontinuous Galerkin finite element method. We propose two different finite element spaces on...
Article
Full-text available
In this paper, we propose a multiscale approach for solving the parameter-dependent elliptic equation with highly heterogeneous coefficients. In particular, we assume that the coefficients have both small scales and high contrast (where the high contrast refers to the large variations in the coefficients). The main idea of our approach is to constr...
Article
In this paper, we study model reduction for multiscale problems in heterogeneous high-contrast media. Our objective is to combine local model reduction techniques that are based on recently introduced spectral multiscale finite element methods (see [19]) with global model reduction methods such as balanced truncation approaches implemented on a coa...
Article
In this paper we discuss robust two-level domain decomposition preconditioners for highly anisotropic heterogeneous multiscale problems. We present a construction of several coarse spaces that employ standard finite element and multiscale basis functions and discuss techniques to reduce the dimensions of coarse spaces without sacrificing the robust...
Article
Full-text available
In this paper, we give an overview of our results [35, 38, 45, 46] from the point of view of coarse-grid multiscale model reduction by highlighting some common issues in coarse-scale approximations and two-level preconditioners. Reduced models discussed in this paper rely on coarse-grid spaces computed by solving local spectral problems. We define...
Article
Full-text available
In this paper, we study robust iterative solvers for finite element systems resulting in approximation of steady-state Richards ’ equation in porous media with highly heterogeneous conductivity fields. It is known that in such cases the contrast, ratio between the highest and lowest values of the conductivity, can adversely affect the performance o...
Article
A discontinuous Galerkin discretization for second order elliptic equations with discontinuous coefficients in 2D is considered. The domain of interest Ω is assumed to be a union of polygonal substructures Ωi of size O(Hi). We allow this substructure decomposition to be geometrically nonconforming. Inside each substructure Ωi, a conforming finite e...
Article
Full-text available
We construct and analyze multigrid methods with nested coarse spaces for second-order elliptic problems with high-contrast multiscale coefficients. The design of the methods utilizes stable multilevel decompositions with a bound that generally grows with the number of levels. To stabilize this growth, in our theory, we use AMLI-cycle multigrid whic...
Chapter
Full-text available
We apply a recently proposed [5] robust overlapping Schwarz method with a certain spectral construction of the coarse space in the setting of element agglomeration algebraic multigrid methods (or agglomeration AMGe) for elliptic problems with high-contrast coefficients. Our goal is to design multilevel iterative methods that converge independent of...
Chapter
Full-text available
We present a new class of coarse spaces for two-level additive Schwarz preconditioners that yield condition number bound independent of the contrast in the media properties. These coarse spaces are an extension of the spaces discussed in [3]. Second order elliptic equations are considered. We present theoretical and numerical results. Detailed desc...
Chapter
Full-text available
We consider the coupling across an interface of a fluid flow and a porous media flow. The differential equations involve Stokes equations in the fluid region,Darcy equations in the porous region, plus a coupling through an interface with Beaver-Joseph-Saffman transmission conditions, see [1, 2, 6, 8]. The discretization consists of P2-P0 finite ele...
Article
In this short note, we discuss variational multiscale methods for solving porous media flows in high-contrast heterogeneous media with rough source terms. Our objective is to separate, as much as possible, subgrid effects induced by the media properties from those due to heterogeneous source terms. For this reason, enriched coarse spaces designed f...