Analysis of incompressible miscible displacement in porous media by characteristics collocation method

Abstract

Miscible displacement of one incompressible fluid by another in a porous medium is modelled by a coupled system of two partial differential equations. The pressure equation is elliptic, whereas the concentration equation is parabolic but normally convection-dominated. In this article, the collocation scheme is used to approximate the pressure equation and another characteristics collocation scheme to treat concentration equation. Existence and uniqueness of solutions of the algorithm are proved. Optimal order error estimate is demonstrated.

Characteristics Collocation Method of Compressible Miscible Displacement with Dispersion

“…We shall check that if n l = , (17) is right at the end of the proof. Similar to the discussion in [6] [8], and the relations (15) (17) and Gronwall lemma, we can get …”

“…[5] proposes the collocation method of two-dimensional variable coefficients elliptic problems. The characteristics collocation scheme for the incompressible flow is given in [6]. The characteristics finite element method for the compressible miscible flow is proved in [7].…”

A fully discrete H 1-Galerkin method with quadrature for nonlinear advection–diffusion–reaction equations

“…We require H r+3 ( ) regularity, mainly due to technical details involved in analysis of the quadrature error in Lemma 3.1. Such extra regularity assumption on the exact solution is essential for analysis of C 1 spline collocation methods that do not involve integrals, but require evaluation of functions at certain quadrature points [7,10,19]. The analysis in [1,2] for finite element Galerkin method with quadrature for a linear biharmonic problem, restricted to the C 1 cubic spline (r = 3) case, requires H r+5 ( ) regularity.…”

“…It is common in quadrature finite element analysis to assume extra regularity [1,2,6,7,10,19]. We require H r+3 ( ) regularity, mainly due to technical details involved in analysis of the quadrature error in Lemma 3.1.…”

“…Trial spaces spanned by C 1 splines are essential for collocation finite element methods, see the survey paper [7] and the recent work [19] and references therein. The finite element Galerkin method for biharmonic problems based on a weak formulation requires an H 2 trial space, see the recent work [1,2] and references therein.…”