Juan Galvis

Victor M. Calo, Y. Efendiev, Juan Galvis et al.

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 snapshot space and a local spectral decomposition, e.g., the use of oversampled regions in order to achieve an efficient model reduction. However, the successful construction of snapshot spaces may be costly if too many local problems need to be solved in order to obtain these spaces. In this paper, we show that this efficiency can be achieved using a moderate quantity of local solutions (or snapshot vectors) with random boundary conditions on oversampled regions with zero forcing. Motivated by the randomized algorithm presented in [19], we consider a snapshot space which consists of harmonic extensions of random boundary conditions defined in a domain larger than the target region. Furthermore, we perform an eigenvalue decomposition in this small space. We study the application of randomized sampling for GMsFEM in conjunction with adaptivity, where local multiscale spaces are adaptively enriched. Convergence analysis is provided. We present representative numerical results to validate the method proposed.

Leonardo A. Poveda, Sebastian Huepo, Victor M. Calo et al.

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 $H^1$ norm.

Juan Galvis, Carlos Vásquez, Luis F. Contreras

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 to its high-contrast permeability. Thus, although we can classically formulate the free-boundary dam problem as in the homogeneous case, a very high resolution will be needed by a standard finite element approximation in order to obtain realistic results that recover the multiscale nature. First, we introduce a fictitious time variable which motivates a suitable time discretization that can be understood as a fixed point iteration to the steady state solution, and we use a duality method to deal with the involved multivalued nonlinear terms. Next, we compute efficient approximations of the pressure and the saturation by using the GMfsFEM method and we can identify the free boundary. More precisely, the GMsGEM method provides numerical results that capture the behavior of the solution due to the variations of the coefficient at the fine-resolution, by just solving linear systems with size proportional to the number of coarse blocks of a coarse-grid (that does not need to be adapted to the variations of the coefficient). Finally, we present illustrative numerical results to validate the proposed methodology.

Leonardo A. Poveda, Sebastian Huepo, Victor M. Calo et al.

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 a background with low conductivity that contains inclusions with different thermal properties. Under this scenario we design a multiscale finite element method to efficiently approximate solutions. The method is based on an asymptotic expansions of the solution in terms of the ratio between the conductivities. The resulting method constructs (locally) finite element basis functions (one for each inclusion). These bases that generate the multiscale finite element space where the approximation of the solution is computed. Numerical experiments show the good performance of the proposed methodology.

Juan Galvis, H. M. Versieux

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 method. The numerical implementation uses harmonically deformed triangulations to inexpensively generate finite element meshes in subdomains. We apply our method to a simplified model for jet flows in pipes and to a simple magnetohydrodynamics model. Finally, we present numerical examples studying the convergence of our scheme.

Djulustan Nikiforov, Leonardo A. Poveda, Dmitry Ammosov et al.

In this work, we extend the meshfree generalized multiscale exponential integration framework introduced in Nikiforov et al. (2025) to the simulation of three-dimensional advection--diffusion problems in heterogeneous and high-contrast media. The proposed approach combines meshfree generalized multiscale finite element methods (GMsFEM) for spatial discretization with exponential integration techniques for time advancement, enabling stable and efficient computations in the presence of stiffness induced by multiscale coefficients and transport effects. We introduce new constructions of multiscale basis functions that incorporate advection either at the snapshot level or within the local spectral problems, improving the approximation properties of the coarse space in advection-dominated regimes. The extension to three-dimensional settings poses additional computational and methodological challenges, including increased complexity in basis construction, higher-dimensional coarse representations, and stronger stiffness effects, which we address within the proposed framework. A series of numerical experiments in three-dimensional domains demonstrates the viability of the method, showing that it preserves accuracy while allowing for significantly larger time steps compared to standard time discretizations. The results highlight the robustness and efficiency of the proposed approach for large-scale multiscale simulations in complex heterogeneous media.

Juan Galvis, O. Andres Cuervo

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 leads to the solution of a coupled system of elliptic equations where the coupling is given by a Kronecker product matrix involving polynomial evaluation matrices. This generalizes the Classical Monte Carlo approximation and Collocation method for approximating functionals of solutions of these equations.

Eduardo Abreu, Ciro Diaz, Juan Galvis et al.

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 conservative fluxes. Furthermore, our approach can be straightforwardly extended to three dimensions. It is also applicable to highly heterogeneous problems where high-order approximation is preferred.

Juan Galvis, Eric Chung, Yalchin Efendiev et al.

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 spectral information of local bilinear forms. We present several approaches to incorporate the spectral information into the coarse problem in order to obtain minimal coarse space dimension. We show that using these coarse spaces, we can obtain a domain decomposition preconditioner with the condition number independent of contrast and small scales. To minimize further the number of iterations until convergence, we use this minimal dimensional coarse spaces in a construction combining them with large overlap local problems that take advantage of the possibility of localizing global fields orthogonal to the coarse space. We obtain a condition number close to 1 for the new method. We discuss possible drawbacks and further extensions.

Michael Presho, Juan Galvis

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-order GMsFEM approximation space. In particular, we impose the finite volume-based restrictions through incorporating a scalar Lagrange multiplier for each mass conservation constraint. This approach can be equivalently viewed as a constraint minimization problem where we minimize the energy functional of the equation restricted to the subspace of functions that satisfy the desired conservation properties. To test the performance of the method we consider equations with heterogeneous permeability coefficients that have high-variation and discontinuities, and couple the resulting fluxes to a two-phase flow model. The increase in accuracy associated with the computation of the GMsFEM pressure solutions is inherited by the flux fields and saturation solutions, and is closely correlated to the size of the reduced-order systems. In particular, the addition of more basis functions to the enriched multiscale space produces solutions that more accurately capture the behavior of the fine scale model. A variety of numerical examples are offered to validate the performance of the method.