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Explicit polynomial preserving trace liftings on a triangle

Ainsworth, M. and Demkowicz, L. (2009) Explicit polynomial preserving trace liftings on a triangle. Mathematische Nachrichten, 282 (5). pp. 640-658. ISSN 0025-584X

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Abstract

We give an explicit formula for a right inverse of the trace operator from the Sobolev space H1(T) on a triangle T to the trace space H1/2(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 L2(T) to H1/2(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.