Motivated by conforming finite element methods for elliptic problems of second order, we analyze the approximation of the gradient of a target function by continuous piecewise polynomial functions over a simplicial mesh. The main result is that the global best approximation error is equivalent to an appropriate sum in terms of the local best approximation errors on elements. Thus, requiring continuity does not downgrade local approximation capability and discontinuous piecewise polynomials essentially do not offer additional approximation power, even for a fixed mesh. This result implies error bounds in terms of piecewise regularity over the whole admissible smoothness range. Moreover, it allows for simple local error functionals in adaptive tree approximation of gradients.
Approximating Gradients with Continuous Piecewise Polynomial Functions / A. Veeser. - In: FOUNDATIONS OF COMPUTATIONAL MATHEMATICS. - ISSN 1615-3375. - (2015). [Epub ahead of print] [10.1007/s10208-015-9262-z]
Approximating Gradients with Continuous Piecewise Polynomial Functions
A. Veeser
2015
Abstract
Motivated by conforming finite element methods for elliptic problems of second order, we analyze the approximation of the gradient of a target function by continuous piecewise polynomial functions over a simplicial mesh. The main result is that the global best approximation error is equivalent to an appropriate sum in terms of the local best approximation errors on elements. Thus, requiring continuity does not downgrade local approximation capability and discontinuous piecewise polynomials essentially do not offer additional approximation power, even for a fixed mesh. This result implies error bounds in terms of piecewise regularity over the whole admissible smoothness range. Moreover, it allows for simple local error functionals in adaptive tree approximation of gradients.File | Dimensione | Formato | |
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