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A posteriori error estimators based on equilibrated fluxes Sarah Cochez Dhondt and Serge Nicaise

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Niveau: Supérieur, Doctorat, Bac+8
A posteriori error estimators based on equilibrated fluxes Sarah Cochez-Dhondt and Serge Nicaise Universite de Valenciennes et du Hainaut Cambresis LAMAV, FR CNRS 2956, Institut des Sciences et Techniques de Valenciennes F-59313 - Valenciennes Cedex 9 France Sarah.Cochez, April 17, 2007 Abstract We consider conforming finite element approximations of reaction-diffusion problems and time-harmonic Maxwell equations. We propose new a posteriori error estimators based on H(div ) and H(curl) conforming finite elements and equilibrated fluxes. It is shown that these estimators give rise to an upper bound where the constant is one up to higher order terms. Lower bounds can also be established with constants depending on the shape regularity of the mesh and the local variation of the coefficients. The reliability and efficiency of the proposed estimator are confirmed by various numerical tests. Key Words equilibrated fluxes, Maxwell equations, a posteriori error estimates. AMS (MOS) subject classification 65N30; 65N15, 65N50, 1 Introduction Among other methods, the finite element method is widely used for the numerical approxima- tion of partial differential equations, see, e.g., [6, 7, 8, 9, 18]. In many engineering applications, adaptive techniques based on a posteriori error estimators have become an indispensable tool to obtain reliable results.

  • based

  • error estimator

  • conforming elements

  • vh ?

  • local neumann problems

  • system

  • consider conforming

  • t?th

  • positive constant

  • bdm elements


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A posteriori error estimators based on equilibrated uxes
Sarah Cochez-Dhondt and Serge Nicaise
Universit¶e de Valenciennes et du Hainaut Cambr¶esis
LAMAV, FR CNRS 2956,
Institut des Sciences et Techniques de Valenciennes
F-59313 - Valenciennes Cedex 9 France
Sarah.Cochez,Serge.Nicaise@univ-valenciennes.fr
April 17, 2007
Abstract
We consider conforming flnite element approximations of reaction-difiusion problems
and time-harmonic Maxwell equations. We propose new a posteriori error estimators
based on H(div) and H(curl) conforming flnite elements and equilibrated uxes. It is
shown that these estimators give rise to an upper bound where the constant is one up to
higher order terms. Lower bounds can also be established with constants depending on
the shape regularity of the mesh and the local variation of the coe–cients. The reliability
and e–ciency of the proposed estimator are conflrmed by various numerical tests.
Key Words equilibrated uxes, Maxwell equations, a posteriori error estimates.
AMS (MOS) subject classiflcation 65N30; 65N15, 65N50,
1
Introduction
Amongothermethods,theflniteelementmethodiswidelyusedforthenumericalapproximationofpartialdifierentialequations,see,e.g.,[6,7,8,9,18]. Inmanyengineeringapplications,
adaptive techniques based on a posteriori error estimators have become an indispensable tool
toobtainreliableresults. Nowadaysthereexistsavastamountofliteratureonlocallydeflned
a posteriori error estimators for problems in structural mechanics or electromagnetism. We
refertothemonographs[2,3,19,26]foragoodoverviewonthistopic. Ingeneral,localupper
and lower bounds are established in order to guarantee the reliability and the e–ciency of
the proposed estimator. Most of the existing approaches involve constants depending on the
shape regularity of the elements and/or of the jumps in the coe–cients; but these
dependenciesareoftennotgiven. Onlyafewnumberofapproachesgivesrisetoestimateswithexplicit
constants, see, e.g., [2, 6, 15, 17, 20, 24]. For Maxwell’s system, only relatively few results
exist. Difierent well established approaches, for the Laplace operator, have been generalized
andadaptedtothisspecialsituation. Residualtypeerrorestimatorswhichmeasurethejump
ofthediscrete uxhavebeenconsideredin[5,10,18,23,25]; hierarchicalerrorestimatorse.g.
in [4], and estimators based on the solution of local problems have been introduced in [13].
Here we use an approach based on equilibrated uxes and H(div)- or H(curl)-conforming
elements. Similar ideas can be found, e.g., in [6, 17, 24]. For an overview on equilibration
1techniques, we refer to [2, 15]. For reaction-difiusion problems, in contrast to [6], we flrst
deflne on the edges an equilibrated ux and then a H(div)-conforming element being
locally conservative by construction. In [6], the authors directly compute suitable conforming
elements by solving local Neumann problems. On the contrary for Maxwell’s system the
construction of equilibrated uxes seems to be impossible and therefore we use the construction
from [6]. In both cases, the error estimator is locally deflned and yields, up to higher order
terms, an upper bound with constant one for the discretization error. We note that our error
estimators are made for partial difierential equations with zero order terms, and the upper
bound one is still valid in this more general situation. Special care is required by the lower
orderterms. InthecaseofMaxwell’sequations,wehavetointroduceasecondapproximation
that takes into account the non-fulfllment of the divergence constraint of the flnite element
approximation. This second approximation has not to be introduced if the zero order term
is not present. Finally lower bounds are proved, moreover for reaction-difiusion problems, we
trace the dependency of the constants with respect to the variation of the coe–cients for all
proposed estimators. For Maxwell’s system this dependency is partially given.
The outline of the paper is as follows: We recall, in Section 2, the scalar reaction-difiusion
problem and its numerical approximation. Section 3 is devoted to the introduction of the
locally deflned error estimators based on Raviart{Thomas or Brezzi{Douglas{Marini (BDM)
elements and the proofs of the upper and lower bounds. The upper bound directly follows
from the construction of the estimators, while the proof of the lower bound relies on suitable
norm equivalences and some properties of the equilibrated uxes. Finally in Section 4, we
treat the time-harmonic Maxwell equations. For both problem classes, some numerical tests
are presented that conflrm the reliability and e–ciency of our error estimators.
2 The two-dimensional reaction-difiusion equation
2Let › be a bounded domain ofR and ¡ its polygonal boundary. We consider the following
elliptic second order boundary value problem with homogeneous mixed boundary conditions:
¡div(aru)+u = f in ›;
u = 0 on ¡ ; (1)D
aru¢n = 0 on ¡ ;N
„ „where ¡=¡ [¡ and ¡ \¡ =;.D N D N
Inthesequel, wesupposethataispiecewiseconstant, namelyweassumethatthereexists
a partition P of › into a flnite set of Lipschitz polygonal domains › ;¢¢¢;› such that, on1 J
each › , a = a where a is a positive constant. For simplicity of notation, we assume thatj j j
¡ hasanon-vanishingmeasure. Thevariationalformulationof(1)involvesthebilinearformD
Z
B(u;v)= (aru¢rv+uv):

2 1 1Given f 2 L (›), the weak formulation consists in flnding u2 H (›) := fu2 H (›) :D
u=0 on ¡ g such thatD
Z
1B(u;v)=(f;v)= fv; 8v2H (›): (2)D

2