{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T18:39:05Z","timestamp":1776796745955,"version":"3.51.2"},"reference-count":55,"publisher":"American Mathematical Society (AMS)","issue":"285","license":[{"start":{"date-parts":[[2014,6,28]],"date-time":"2014-06-28T00:00:00Z","timestamp":1403913600000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>In this paper we derive an a posteriori error estimate for the numerical approximation of the solution of a system modeling the flow of two incompressible and immiscible fluids in a porous medium. We take into account the capillary pressure, which leads to a coupled system of two equations: parabolic and elliptic. The parabolic equation may become degenerate, i.e., the nonlinear diffusion coefficient may vanish over regions that are not known a priori. We first show that, under appropriate assumptions, the energy-type norm differences between the exact and the approximate nonwetting phase saturations, the global pressures, and the Kirchhoff transforms of the nonwetting phase saturations can be bounded by the dual norm of the residuals. We then bound the dual norm of the residuals by fully computable a posteriori estimators. Our analysis covers a large class of conforming, vertex-centered finite volume-type discretizations with fully implicit time stepping. As an example, we focus here on two approaches: a \u201cmathematical\u201d scheme derived from the weak formulation, and a phase-by-phase upstream weighting \u201cengineering\u201d scheme. Finally, we show how the different error components, namely the space discretization error, the time discretization error, the linearization error, the algebraic solver error, and the quadrature error can be distinguished and used for making the calculations efficient.<\/p>","DOI":"10.1090\/s0025-5718-2013-02723-8","type":"journal-article","created":{"date-parts":[[2013,6,28]],"date-time":"2013-06-28T11:40:01Z","timestamp":1372419601000},"page":"153-188","source":"Crossref","is-referenced-by-count":43,"title":["An a posteriori error estimate for vertex-centered finite volume discretizations of immiscible incompressible two-phase flow"],"prefix":"10.1090","volume":"83","author":[{"given":"Cl\u00e9ment","family":"Canc\u00e8s","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Iuliu","family":"Pop","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Martin","family":"Vohral\u00edk","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2013,6,28]]},"reference":[{"key":"1","series-title":"Pure and Applied Mathematics (New York)","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1002\/9781118032824","volume-title":"A posteriori error estimation in finite element analysis","author":"Ainsworth, Mark","year":"2000","ISBN":"https:\/\/id.crossref.org\/isbn\/047129411X"},{"issue":"254","key":"2","doi-asserted-by":"publisher","first-page":"511","DOI":"10.1090\/S0025-5718-05-01800-4","article-title":"A posteriori error estimates for the Crank-Nicolson method for parabolic equations","volume":"75","author":"Akrivis, Georgios","year":"2006","journal-title":"Math. 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