# American Institute of Mathematical Sciences

2013, 3(2): 327-345. doi: 10.3934/naco.2013.3.327

## An adaptive wavelet method and its analysis for parabolic equations

 1 Department of Mathematics and Statistics, York University, Toronto, Ontario, M3J 1P3, Canada

Received  April 2012 Revised  March 2013 Published  April 2013

In this paper, we analyze an adaptive wavelet method with variable time step sizes and space refinement for parabolic equations. The advantages of multi-resolution wavelet processes combined with certain equivalences involving weighted sequence norms of wavelet coefficients allow us to set up an efficient adaptive algorithm producing locally refined spaces for each time step. Reliable and efficient a posteriori error estimate is derived, which assesses the discretization error with respect to a given quantity of physical interest. The influence of the time and space discretization errors is separated into different error indicators. We prove that the proposed adaptive wavelet algorithm terminates in a finite number of iterations for any given accuracy.
Citation: Qiang Guo, Dong Liang. An adaptive wavelet method and its analysis for parabolic equations. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 327-345. doi: 10.3934/naco.2013.3.327
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