2011
DOI: 10.1016/j.crma.2011.09.005
|View full text |Cite
|
Sign up to set email alerts
|

The effect of numerical integration in the finite element method for nonmonotone nonlinear elliptic problems with application to numerical homogenization methods

Abstract: To cite this version:Assyr Abdulle, Gilles Vilmart. The effect of numerical integration in the finite element method for nonmonotone nonlinear elliptic problems with application to numerical homogenization methods. Comptes rendus hebdomadaires des séances de l'Académie des sciences, Elsevier, 2011Elsevier, , 349, pp.1041Elsevier, -1046Elsevier, . <10.1016Elsevier, /j.crma.2011.005>. Numerical AnalysisThe effect of numerical integration in the finite element method for nonmonotone nonlinear e… Show more

Help me understand this report

Search citation statements

Order By: Relevance

Paper Sections

Select...
3
2

Citation Types

0
10
0

Year Published

2011
2011
2015
2015

Publication Types

Select...
5

Relationship

4
1

Authors

Journals

citations
Cited by 8 publications
(10 citation statements)
references
References 16 publications
0
10
0
Order By: Relevance
“…A priori error estimates similar to theorem 4.1 have been obtained for parabolic problems of the type (2.2) in [43], for nonlinear elliptic problems of the type (2.4) in [40,41] and for wave problems of type (2.3) in [42].…”
Section: Where C Is Independent Of H H and εmentioning
confidence: 90%
See 1 more Smart Citation
“…A priori error estimates similar to theorem 4.1 have been obtained for parabolic problems of the type (2.2) in [43], for nonlinear elliptic problems of the type (2.4) in [40,41] and for wave problems of type (2.3) in [42].…”
Section: Where C Is Independent Of H H and εmentioning
confidence: 90%
“…For the nonlinear problem (2.4), a similar procedure can be used [40,41], thanks to the results in [46, section 3.4.2]. There it is shown that any corrector u 1,ε for the linear problem obtained from (2.4) by replacing a ε (x, u ε (x)) with a ε (x, u 0 (x)), where u 0 is the solution of the corresponding homogenization problem, is also a corrector for the solution u ε of the nonlinear problem (2.4) and that…”
Section: Where C Is Independent Of H H and εmentioning
confidence: 99%
“…Notice that we have used the symmetry in the above proof. This proof is however valid without symmetry (following ideas in [12] or [32,Lemma 4.6], [33]). Next we derive an a posteriori estimator, which allows to control the accuracy of our output of interest (the numerically homogenized tensor).…”
Section: Model Reductionmentioning
confidence: 98%
“…Indeed, such methods are based on a macroscopic solver whose bilinear form is obtained by numerical quadrature, with data recovered by microscopic solvers defined on sampling domains at the quadrature nodes [1,2,15]. Convergence rates for FEMs with numerical quadrature are thus essential in the analysis of numerical homogenization methods and the a priori error bounds derived in this paper allow to use an approach similar to the linear case for the analysis of nonlinear homogenization problems [3,4].…”
Section: Introductionmentioning
confidence: 99%