Rates of Convergence of Approximate Solutions of Parabolic Initial-Boundary Value Problems
| Autor: | C. S. Caldwell |
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| Rok vydání: | 1974 |
| Předmět: | |
| Zdroj: | SIAM Journal on Numerical Analysis. 11:1121-1135 |
| ISSN: | 1095-7170 0036-1429 |
| DOI: | 10.1137/0711085 |
| Popis: | This paper presents error estimates for finite difference approximate solutions of initial-boundary problems for parabolic partial differential equations. Methods having matrices of essentially positive type are considered using many techniques from Bramble, Hubbard and Thomee [2] where elliptic partial differential equations were examined.It will be shown that if the approximation has “accuracy” $\nu $, the right-hand side of the differential equation f has “smoothness” $\lambda $ and the initial and boundary data $\varphi $ and $\Phi $ have “smoothness” $\mu $, then letting$| \cdot |_A^{(r)} $ denote a Holder-type norm of order r in the set A we have for $\mu \ne \nu $ that the maximum error is bounded by \[ C\{ h^{\min (\lambda ,v)} | f |_\mathcal{L}^{(\lambda )} + h^{\min (\mu ,\nu )} | {\varphi ,\Phi } |_{\mathcal{R} \times \mathcal{L}}^{(\mu )} \} \] and for $\mu = \nu $ and $f \equiv 0$, this error is dominated by \[ Ch^\nu \left( {\ln \frac{1}{h}} \right)| {(\varphi ,\Phi )} |_{\mathcal{R} \times ... |
| Databáze: | OpenAIRE |
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