On the sharpness of a superconvergence estimate in connection with one-dimensional Galerkin methods
<p/> <p>The present paper studies some aspects of approximation theory in the context of one-dimensional Galerkin methods. The phenomenon of superconvergence at the knots is well-known. Indeed, for smooth solutions the rate of convergence at these points is <inline-formula><grap...
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| Format: | Article |
| Language: | English |
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SpringerOpen
1999-01-01
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| Series: | Journal of Inequalities and Applications |
| Subjects: | |
| Online Access: | http://www.journalofinequalitiesandapplications.com/content/3/351053 |
| Summary: | <p/> <p>The present paper studies some aspects of approximation theory in the context of one-dimensional Galerkin methods. The phenomenon of superconvergence at the knots is well-known. Indeed, for smooth solutions the rate of convergence at these points is <inline-formula><graphic file="1029-242X-1999-351053-i1.gif"/></inline-formula> instead of <inline-formula><graphic file="1029-242X-1999-351053-i2.gif"/></inline-formula>, where <inline-formula><graphic file="1029-242X-1999-351053-i3.gif"/></inline-formula> is the degree of the finite element space. In order to achieve a corresponding result for less smooth functions, we apply K-functional techniques to a Jackson-type inequality and estimate the relevant error by a modulus of continuity. Furthermore, this error estimate requires no additional assumptions on the solution, and it turns out that it is sharp in connection with general Lipschitz classes. The proof of the sharpness is based upon a quantitative extension of the uniform boundedness principle in connection with some ideas of Douglas and Dupont [<it>Numer. Math.</it> 22] (1974) 99–109. Here it is crucial to design a sequence of test functions such that a Jackson–Bernstein-type inequality and a resonance condition are satisfied simultaneously.</p> |
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| ISSN: | 1025-5834 1029-242X |