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Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Some optimal error estimates for piecewise linear finite element approximations

Authors: Rolf Rannacher and Ridgway Scott
Journal: Math. Comp. 38 (1982), 437-445
MSC: Primary 65N30
MathSciNet review: 645661
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Abstract: It is shown that the Ritz projection onto spaces of piecewise linear finite elements is bounded in the Sobolev space, $ \hat{W}_p^1$, for $ 2 \leqslant p \leqslant \infty $. This implies that for functions in $ \hat{W}_p^1 \cap W_p^2$ the error in approximation behaves like $ O(h)$ in $ W_p^1$, for $ 2 \leqslant p \leqslant \infty $, and like $ O({h^2})$ in $ {L_p}$, for $ 2 \leqslant p < \infty $. In all these cases the additional logarithmic factor previously included in error estimates for linear finite elements does not occur.

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Keywords: Maximum norm estimates, finite element methods
Article copyright: © Copyright 1982 American Mathematical Society