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title:
On the convergence analysis of a spatially adaptive Rothe method II |
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name:
Doehring |
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first name:
Nicolas
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location/conference:
SPP-JT12
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WWW-link:
http://www.dfg-spp1324.de/download/preprints/preprint124.pdf |
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PRESENTATION-link:
http://www.dfg-spp1324.de/nuhagtools/event_NEW/dateien/SPP-JT12/talks/Doehring-jt12.pdf |
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abstract:
This talk is about the convergence analysis of the horizontal method of lines for (deterministic or stochastic) parabolic evolution equations. We use uniform discretizations in time and nonuniform (adaptive) discretizations in space. The space discretization methods are assumed to converge up to a given tolerance $\epsilon$ when applied to the resulting elliptic subproblems. We investigate how the tolerances $\epsilon$ in each time step have to be tuned so that the overall scheme converges with the same order as in the case of exact evaluations of the elliptic subproblems. We show that the analysis can be applied to rather general classes of deterministic parabolic problems and arbitrary $S$-stage discretization schemes. We present a detailed analysis of the case of adaptive wavelet discretizations in space. Using concepts from regularity theory for partial differential equations and from nonlinear approximation theory, we determine an upper bound for the degrees of freedom for the overall scheme that are needed to adaptively approximate the solution up to a prescribed tolerance. |
