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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show all references

References:
[1]

Journal of Computational Physics, 214 (2006), 829-857. doi: 10.1016/j.jcp.2005.10.009.  Google Scholar

[2]

Mathematical Modelling and Numerical Analysis, 26 (1992), 793-834.  Google Scholar

[3]

SIAM Journal on Scientific Computing, 23 (2001), 910-939. doi: 10.1137/S1064827599365501.  Google Scholar

[4]

Computers and Chemical Engineering, 27 (2003), 131-142. doi: 10.1016/S0098-1354(02)00165-5.  Google Scholar

[5]

Applied and Computational Harmonic Analysis, 6 (1999), 1-52. doi: 10.1006/acha.1997.0242.  Google Scholar

[6]

Numerische Mathematik, 103 (2006), 367-392. doi: 10.1007/s00211-006-0677-y.  Google Scholar

[7]

Mathematics of Computation, 73 (2004), 1167-1193. doi: 10.1090/S0025-5718-04-01634-5.  Google Scholar

[8]

Elsevier, 2003. Google Scholar

[9]

Mathematics of Computation, 70 (2001), 27-75. doi: 10.1090/S0025-5718-00-01252-7.  Google Scholar

[10]

Foundations of Computational Mathematics, 2 (2002), 203-245. doi: 10.1007/s102080010027.  Google Scholar

[11]

Communications on Pure and Applied Mathematics, 45 (1992), 485-560. doi: 10.1002/cpa.3160450502.  Google Scholar

[12]

Applied and Computational Harmonic Analysis, 1 (1993), 54-81. doi: 10.1006/acha.1993.1005.  Google Scholar

[13]

SIAM Journal on Scientific Computing, 21 (1999), 1006-1026. doi: 10.1137/S1064827597330613.  Google Scholar

[14]

Applied Numerical Mathematics, 23 (1997), 21-47. doi: 10.1016/S0168-9274(96)00060-8.  Google Scholar

[15]

SIAM Journal on Control and Optimization, 43 (2005), 1640-1675. doi: 10.1137/S0363012902419199.  Google Scholar

[16]

Applied and Computational Harmonic Analysis, 6 (1999), 132-196. doi: 10.1006/acha.1998.0247.  Google Scholar

[17]

Advances in Computational Mathematics, 1 (1993), 259-335. doi: 10.1007/BF02072014.  Google Scholar

[18]

Communications on Pure and Applied Mathematics, 41 (1988), 909-996. doi: 10.1002/cpa.3160410705.  Google Scholar

[19]

SIAM Philadelphia, 1992. doi: 10.1137/1.9781611970104.  Google Scholar

[20]

Tech. Rep., NASA Contractor Report 187480, ICASE Report 90-83, NASA Langley Research Center, Hampton VA 23665-5225 (1990). Google Scholar

[21]

Journal of Aerosol Science, 39 (2008), 467-487. doi: 10.1016/j.jaerosci.2008.01.005.  Google Scholar

[22]

SIAM Journal on Numerical Analysis, 38 (2000), 466-488. doi: 10.1137/S0036142999360044.  Google Scholar

[23]

Journal of Computational Physics, 188 (2003), 493-523.  Google Scholar

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