Two Families Of H(div) Mixed Finite Elements On Quadrilaterals Of Minimal Dimension

dc.creatorArbogast
dc.creatorTodd; Correa
dc.creatorMaicon R.
dc.date2016
dc.date2017-11-13T11:34:13Z
dc.date2017-11-13T11:34:13Z
dc.date.accessioned2018-03-29T05:48:33Z
dc.date.available2018-03-29T05:48:33Z
dc.identifierSiam Journal On Numerical Analysis. Siam Publications, v. 54, p. 3332 - 3356, 2016.
dc.identifier0036-1429
dc.identifier1095-7170
dc.identifierWOS:000391855200007
dc.identifier10.1137/15M1013705
dc.identifierhttp://epubs.siam.org/doi/abs/10.1137/15M1013705
dc.identifierhttp://repositorio.unicamp.br/jspui/handle/REPOSIP/326388
dc.identifier.urihttp://repositorioslatinoamericanos.uchile.cl/handle/2250/1363394
dc.descriptionFundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)
dc.descriptionWe develop two families of mixed finite elements on quadrilateral meshes for approximating (u,p) solving a second order elliptic equation in mixed form. Standard Raviart-Thomas (RT) and Brezzi-Douglas-Marini (BDM) elements are defined on rectangles and extended to quadrilaterals using the Piola transform, which are well-known to lose optimal approximation of del.u. Arnold-Boffi-Falk spaces rectify the problem by increasing the dimension of RT, so that approximation is maintained after Piola mapping. Our two families of finite elements are uniformly inf-sup stable, achieve optimal rates of convergence, and have minimal dimension. The elements for u are constructed from vector polynomials defined directly on the quadrilaterals, rather than being transformed from a reference rectangle by the Piola mapping, and then supplemented by two (one for the lowest order) basis functions that are Piola mapped. One family has full H(div)-approximation (u, p, and del.u are approximated to the same order like RT) and the other has reduced H(div)-approximation (p and del.u are approximated to one less power like BDM). The two families are identical except for inclusion of a minimal set of vector and scalar polynomials needed for higher order approximation of del.u and p, and thereby we clarify and unify the treatment of finite element approximation between these two classes. The key result is a Helmholtz-like decomposition of vector polynomials, which explains precisely how a divergence is approximated locally. We develop an implementable local basis and present numerical results confirming the theory.
dc.description54
dc.description6
dc.description3332
dc.description3356
dc.descriptionU.S. National Science Foundation [DMS-1418752]
dc.descriptionSao Paulo Research Foundation (FAPESP) [2013/19021-8]
dc.descriptionFundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)
dc.languageEnglish
dc.publisherSIAM Publications
dc.publisherPhiladelphia
dc.relationSiam Journal on Numerical Analysis
dc.rightsaberto
dc.sourceWOS
dc.subjectSecond Order Elliptic Equation
dc.subjectMixed Methods
dc.subjectDivergence Approximation
dc.subjectFull H(div)-approximation
dc.subjectReduced H(div)-approximation
dc.subjectInf-sup Stable
dc.titleTwo Families Of H(div) Mixed Finite Elements On Quadrilaterals Of Minimal Dimension
dc.typeArtículos de revistas


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