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Publication# Mixed finite element -- primal finite volume element discretization of multicontinuum models

Abstract

This paper is devoted to the mathematical modelling and numerical simulation of basic mechanisms that drive multicontinuum systems in two- and three-dimensional porous media. We state a general mathematical model within the framework of mixture theory, capable to describe and predict flow patterns occurring in permeable media, a problem exhibiting high degrees of heterogeneity and large disparities in physical scales. The model equations result in a strongly coupled and nonlinear system of partial differential equations (PDE) that are written in terms of phase and barycentric mixture velocities, phase pressure, and saturation. We construct accurate, robust and reliable hybrid (or combined) mixed finite element -- primal finite volume-element methods for the discretization of the underlying equations on unstructured meshes. These schemes rely on mixed Brezzi-Douglas-Marini approximations of phase and total velocities, on piecewise constant elements for the approximation of phase or total pressures, and on a primal formulation using discontinuous finite volume elements defined on a dual diamond mesh to approximate scalar mixture constituents of interest (such as volume fraction, total density, saturation, etc.). A second order backward difference formula is employed in the approximation of time derivatives. Several numerical test cases are presented and discussed in detail, illustrating the validity of the approach proposed herein along with the accuracy of the mixed--primal method. We comment as well on the applicability of the model and numerical scheme to the study of other related systems of wide interest.

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Numerical methods for ordinary differential equations

Numerical methods for ordinary differential equations are methods used to find numerical approximations to the solutions of ordinary differential equations (ODEs). Their use is also known as "numerical integration", although this term can also refer to the computation of integrals. Many differential equations cannot be solved exactly. For practical purposes, however – such as in engineering – a numeric approximation to the solution is often sufficient. The algorithms studied here can be used to compute such an approximation.

Numerical analysis

Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods that attempt at finding approximate solutions of problems rather than the exact ones. Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences, medicine, business and even the arts.

Finite difference

A finite difference is a mathematical expression of the form f (x + b) − f (x + a). If a finite difference is divided by b − a, one gets a difference quotient. The approximation of derivatives by finite differences plays a central role in finite difference methods for the numerical solution of differential equations, especially boundary value problems. The difference operator, commonly denoted is the operator that maps a function f to the function defined by A difference equation is a functional equation that involves the finite difference operator in the same way as a differential equation involves derivatives.

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