# Article

 Title: Continuous-time finite element analysis of multiphase flow in groundwater hydrology (English) Author: Chen, Zhangxin Author: Espedal, Magne Author: Ewing, Richard E. Language: English Journal: Applications of Mathematics ISSN: 0862-7940 (print) ISSN: 1572-9109 (online) Volume: 40 Issue: 3 Year: 1995 Pages: 203-226 Summary lang: English . Category: math . Summary: 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. (English) Keyword: mixed method Keyword: finite element Keyword: compressible flow Keyword: porous media Keyword: error estimate Keyword: air-water system MSC: 65M60 MSC: 65N30 MSC: 76M10 MSC: 76S05 idZBL: Zbl 0847.76030 idMR: MR1332314 DOI: 10.21136/AM.1995.134291 . 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