arXiv subjects

J. Galvis

Publications and source records attributed to J. Galvis.

5 recordsLinked to original sources

arXiv2023

Multicontinuum homogenization. General theory and applications

In this paper, we discuss a general framework for multicontinuum homogenization. Multicontinuum models are widely used in many applications and some derivations for these models are established. In these models, several macroscopic variables at each macroscale point are defined and the resulting multicontinuum equations are formulated. In this paper, we propose a general formulation and associated ingredients that allow performing multicontinuum homogenization. Our derivation consists of several main parts. In the first part, we propose a general expansion, where the solution is expressed via the product of multiple macro variables and associated cell problems. The second part consists of formulating the cell problems. The cell problems are formulated as saddle point problems with constraints for each continua. Defining the continua via test functions, we set the constraints as an integral representation. Finally, substituting the expansion to the original system, we obtain multicontinuum systems. We present an application to the mixed formulation of elliptic equations. This is a challenging system as the system does not have symmetry. We discuss the local problems and various macroscale representations for the solution and its gradient. Using various order approximations, one can obtain different systems of equations. We discuss the applicability of multicontinuum homogenization and relate this to high contrast in the cell problem. Numerical results are presented.

math.NA↗

arXiv2021

A note on averaging prediction accuracy, Green's functions and other kernels

We present the mathematical context of the predictive accuracy index and then introduce the definition of integral average transform. We establish the relation of our definition with two variables kernels $K({\bf y},{\bf x})$. As an example of an application we show that integrating against the fundamental solution of the Laplace operator, that is, solving the Poisson equation, can be re-interpreted as an integral of averages of the forcing term over balls. As a result, we obtained a novel integral representation of the solution of the Poisson equation. 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.

cs.CR↗

arXiv2021

A porous media fracture model based on homogenization theory

A novel regularized fracture model for crack propagation in porous media is proposed. Our model is obtained through homogenization theory and formal asymptotic expansions. We start with a regularized quasi-static fracture model posed in a periodically perforated domain obtained by periodic extension of a re-scaled unit cell with a hole. This setup allows us to write two separated minimality conditions for the primary (displacement) and secondary variables plus a balance of energy relation. Then we apply the usual asymptotic expansion matching to deduce limit relations when the re-scaling parameter of the unit cells vanishes. By introducing cell problems solutions and a homogenized tensor we can recast the obtained relations into a novel model for crack propagation in porous media. The proposed model can be interpreted as a regularized quasi-static fracture model for porous media. This model yields two separated (homogenized) minimality conditions for the primary and secondary variables and a balance of homogenized energy relation.

math.AP↗

arXiv2013

Generalized Multiscale Finite Element Method. Symmetric Interior Penalty Coupling

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 the coarse mesh. The first space is based on a local eigenvalue problem that uses a weighted $L_2-$norm for computing the "mass" matrix. The second space is generated by amending the eigenvalue problem of the first case with a term related to the penalty. The third choice is based on generation of a large space of snapshots and subsequent selection of a subspace of a reduced dimension. The approximation with these spaces is based on the discontinuous Galerkin finite element method framework. We investigate the stability and derive error estimates for the methods and further experimentally study their performance on a representative number of numerical examples.

math.NA↗

arXiv2011

Robust Domain Decomposition Preconditioners for Abstract Symmetric Positive Definite Bilinear Forms

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 (pressure) equation and the stream function formulation of its mixed form, Stokes' and Brinkman's equations. The constant in the corresponding abstract energy estimate is shown to be robust with respect to mesh parameters as well as the contrast, which is defined as the ratio of high and low values of the conductivity (or permeability). The derived stable decomposition allows to construct additive overlapping Schwarz iterative methods with condition numbers uniformly bounded with respect to the contrast and mesh parameters. The coarse spaces are obtained by patching together the eigenfunctions corresponding to the smallest eigenvalues of certain local problems. A detailed analysis of the abstract setting is provided. The proposed decomposition builds on a method of Efendiev and Galvis, Multiscale Model. Simul., 8 (2010), pp. 1461--1483, developed for second order scalar elliptic problems with high contrast. Applications to the finite element discretizations of the second order elliptic problem in Galerkin and mixed formulation, the Stokes equations, and Brinkman's problem are presented. A number of numerical experiments for these problems in two spatial dimensions are provided.

math.NA↗