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Explicit polynomial preserving trace liftings on a triangle
Author(s) -
Ainsworth Mark,
Demkowicz Leszek
Publication year - 2009
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200610762
Subject(s) - mathematics , sobolev space , trace (psycholinguistics) , trace operator , polynomial , piecewise , inverse , differential operator , bounded function , boundary (topology) , degree (music) , operator (biology) , pure mathematics , order (exchange) , combinatorics , mathematical analysis , free boundary problem , geometry , finance , economics , elliptic boundary value problem , philosophy , linguistics , physics , biochemistry , chemistry , repressor , acoustics , transcription factor , gene
We give an explicit formula for a right inverse of the trace operator from the Sobolev space H 1 ( T ) on a triangle T to the trace space H 1/2 (∂ T ) on the boundary. The lifting preserves polynomials in the sense that if the boundary data are piecewise polynomial of degree N , then the lifting is a polynomial of total degree at most N and the lifting is shown to be uniformly stable independently of the polynomial order. Moreover, the same operator is shown to provide a uniformly stable lifting from L 2 (∂ T ) to H 1/2 ( T ). Finally, the lifting is used to construct a uniformly bounded right inverse for the normal trace operator from the space H (div; T ) to H –1/2 (∂ T ) which also preserves polynomials. Applications to the analysis of high order numerical methods for partial differential equations are indicated (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)