Continuous-time finite element analysis of multiphase flow in groundwater hydrology

Zhangxin Chen; Magne Espedal; Richard E. Ewing

Applications of Mathematics (1995)

  • Volume: 40, Issue: 3, page 203-226
  • ISSN: 0862-7940

Abstract

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A nonlinear differential system for describing an air-water system in groundwater hydrology is given. The system is written in a fractional flow formulation, i.e., in terms of a saturation and a global pressure. A continuous-time version of the finite element method is developed and analyzed for the approximation of the saturation and pressure. The saturation equation is treated by a Galerkin finite element method, while the pressure equation is treated by a mixed finite element method. The analysis is carried out first for the case where the capillary diffusion coefficient is assumed to be uniformly positive, and is then extended to a degenerate case where the diffusion coefficient can be zero. It is shown that error estimates of optimal order in the L 2 -norm and almost optimal order in the L -norm can be obtained in the nondegenerate case. In the degenerate case we consider a regularization of the saturation equation by perturbing the diffusion coefficient. The norm of error estimates depends on the severity of the degeneracy in diffusivity, with almost optimal order convergence for non-severe degeneracy. Existence and uniqueness of the approximate solution is also proven.

How to cite

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Chen, Zhangxin, Espedal, Magne, and Ewing, Richard E.. "Continuous-time finite element analysis of multiphase flow in groundwater hydrology." Applications of Mathematics 40.3 (1995): 203-226. <http://eudml.org/doc/32916>.

@article{Chen1995,
abstract = {A nonlinear differential system for describing an air-water system in groundwater hydrology is given. The system is written in a fractional flow formulation, i.e., in terms of a saturation and a global pressure. A continuous-time version of the finite element method is developed and analyzed for the approximation of the saturation and pressure. The saturation equation is treated by a Galerkin finite element method, while the pressure equation is treated by a mixed finite element method. The analysis is carried out first for the case where the capillary diffusion coefficient is assumed to be uniformly positive, and is then extended to a degenerate case where the diffusion coefficient can be zero. It is shown that error estimates of optimal order in the $L^2$-norm and almost optimal order in the $L^\infty $-norm can be obtained in the nondegenerate case. In the degenerate case we consider a regularization of the saturation equation by perturbing the diffusion coefficient. The norm of error estimates depends on the severity of the degeneracy in diffusivity, with almost optimal order convergence for non-severe degeneracy. Existence and uniqueness of the approximate solution is also proven.},
author = {Chen, Zhangxin, Espedal, Magne, Ewing, Richard E.},
journal = {Applications of Mathematics},
keywords = {mixed method; finite element; compressible flow; porous media; error estimate; air-water system; saturation; pressure; Galerkin finite element method; mixed finite element method},
language = {eng},
number = {3},
pages = {203-226},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Continuous-time finite element analysis of multiphase flow in groundwater hydrology},
url = {http://eudml.org/doc/32916},
volume = {40},
year = {1995},
}

TY - JOUR
AU - Chen, Zhangxin
AU - Espedal, Magne
AU - Ewing, Richard E.
TI - Continuous-time finite element analysis of multiphase flow in groundwater hydrology
JO - Applications of Mathematics
PY - 1995
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 40
IS - 3
SP - 203
EP - 226
AB - A nonlinear differential system for describing an air-water system in groundwater hydrology is given. The system is written in a fractional flow formulation, i.e., in terms of a saturation and a global pressure. A continuous-time version of the finite element method is developed and analyzed for the approximation of the saturation and pressure. The saturation equation is treated by a Galerkin finite element method, while the pressure equation is treated by a mixed finite element method. The analysis is carried out first for the case where the capillary diffusion coefficient is assumed to be uniformly positive, and is then extended to a degenerate case where the diffusion coefficient can be zero. It is shown that error estimates of optimal order in the $L^2$-norm and almost optimal order in the $L^\infty $-norm can be obtained in the nondegenerate case. In the degenerate case we consider a regularization of the saturation equation by perturbing the diffusion coefficient. The norm of error estimates depends on the severity of the degeneracy in diffusivity, with almost optimal order convergence for non-severe degeneracy. Existence and uniqueness of the approximate solution is also proven.
LA - eng
KW - mixed method; finite element; compressible flow; porous media; error estimate; air-water system; saturation; pressure; Galerkin finite element method; mixed finite element method
UR - http://eudml.org/doc/32916
ER -

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