Numerical results on the residual error indicator for a contaminant transport model

João Paulo Martins dos Santos, Alessandro Firmiano, Edson Wendland


In this article, residual indicators were used to characterize the quality of the numerical solution of the advection-diffusion-reaction equation in a saturated porous medium. Both large and small advection regimes were considered. The small advection was exemplified by a problem with constant data, while non-constant data was considered for the large advection regime. In this case, residual quantities associated with the data must be incorporated into the residual estimates related to the spatial approximation and an auxiliary problem must be solved for the correct obtainment of the temporal estimates. The presentation of the residual indicators as a surface on the finite element mesh provides a detailed view of the regions that need refinement, allows to infer the effect of each estimate on the composition of the global estimator and, in addition, allows to follow the evolution of the residual surfaces as the contaminant front advances in the simulation process. In turn, the numerical values of the indicators allow to delimit the elements that will be refined, to compare the magnitude of the contributions among themselves, between different meshes and a better understanding of the composition of the global estimates.


Advection-Dispersion-Reaction; $\theta$-Scheme; Finite Element; Residual error.

Full Text:





R.~E. Ewing, ``A posteriori error estimation,'' {em Computer Methods in

Applied Mechanics and Engineering}, vol.~82, no.~1-3, pp.~59--72, 1990.


R.~Verf"{u}rth, ``A posteriori error estimates for linear parabolic

equations,'' 2004.


D.~Praetorius, E.~Weinmuller, and P.~Wissgott, ``A space-time adaptive

algorithm for linear parabolic problems,'' asc report 07/2008, Institute for

Analysis and Scientific Computing Vienna University of Technology — TU

Wien, 2008.

newblock available at ISBN 978-3-902627-00-1.


R.~VERF"{U}RTH, ``Adaptive finite element methods: Lecture notes winter term

/14,'' 2014.


J.~Bear, {em Hydraulics of Groudwater}.

newblock Dover Publications, Inc., 1979.


V.~Batu, {em Applied Flow and Solute Transport Modeling in Aquifers:

Fundamental Principles and Analytical and Numerical Methods}.

newblock CRC Press Taylor, 2006.


``Fenics project documentation. url:''


S.~Brenner and L.~Scott, ``The mathematical theory of finite element methods,''

in {em Texts in Applied Mathematics, v. 15}, New York: Springer-Verlag,


E.~Jones, T.~Oliphant, P.~Peterson, {em et~al.}, ``{SciPy}: Open source

scientific tools for {Python},'' 2001.


S.~van~der Walt, S.~Colbert, and G.~Varoquaux, ``The {NumPy} {Array}: {A}

{Structure} for {Efficient} {Numerical} {Computation},'' {em Computing in

Science Engineering}, vol.~13, pp.~22--30, Mar. 2011.


J.~D. Hunter, ``Matplotlib: A 2d graphics environment,'' {em Computing In

Science & Engineering}, vol.~9, no.~3, pp.~90--95, 2007.


G.~Pinder, ``A galerkin-finite element simulation of groundwater contamination

on long island,'' {em Water Resources Research}, vol.~9, no.~6,

pp.~1657--1669, 1973.


E.~Wexler, ``Chapter b7: Applications of hydraulics analytical solutions for

one-, two- and three-dimensional solute transport in groundwater systems with

uniform flow,'' in {em Techniques of Water Resources Investigations of the

United State Geological Survey, Book 3: Applications to Hydraulics}, Denver,

USA: U.S. Geological Survey, 1992.


J.~Santos, A.~Firmiano, and E.~Wendland, ``Jump dominance on the contaminant

transport residual error estimator,'' {em TEMA - Trends in Applied and

Computational Mathematics}, no.~1, 2014.


C.~Zoppou and J.~Knight, ``Analytical solution of a spatially variable

coefficient advection-diffusion equation in up to three dimensions,'' {em

Applied Mathematical Modelling}, vol.~23, no.~9, pp.~667--685, 1999.


Article Metrics

Metrics Loading ...

Metrics powered by PLOS ALM


  • There are currently no refbacks.

Trends in Computational and Applied Mathematics

A publication of the Brazilian Society of Applied and Computational Mathematics (SBMAC)


Indexed in:



Desenvolvido por:

Logomarca da Lepidus Tecnologia