# 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 and Optimization, 2013, 3 (2) : 327-345. doi: 10.3934/naco.2013.3.327
##### References:
 [1] J. M. Alam, N. K.-R. Kevlahan and O. V. Vasilyev, Simultaneous space-time adaptive wavelet solution of nonlinear parabolic differential equations, Journal of Computational Physics, 214 (2006), 829-857. doi: 10.1016/j.jcp.2005.10.009. [2] E. Bacry, S. Mallat and G. Papanicolaou, A wavelet based space-time adaptive numerical method for partial differential equation, Mathematical Modelling and Numerical Analysis, 26 (1992), 793-834. [3] A. Barinka, T. Barsch, P. Charton, A. Cohen, S. Dahlke, W. Dahmen and K. Urban, Adaptive wavelet schemes for elliptic problems-implementation and numerical experiments, SIAM Journal on Scientific Computing, 23 (2001), 910-939. doi: 10.1137/S1064827599365501. [4] A. Bindal, J. G. Khinast and M. G. Ierapetritou, Adaptive multiscale solution of dynamical systems in chemical processes using wavelets, Computers and Chemical Engineering, 27 (2003), 131-142. doi: 10.1016/S0098-1354(02)00165-5. [5] C. Canuto, A. Tabacco and K. Urban, The wavelet element method - Part I. Construction and analysis, Applied and Computational Harmonic Analysis, 6 (1999), 1-52. doi: 10.1006/acha.1997.0242. [6] J. M. Cascón, L. Ferragut and M. I. Asensio, Space-time adaptive algorithm for the mixed parabolic problem, Numerische Mathematik, 103 (2006), 367-392. doi: 10.1007/s00211-006-0677-y. [7] Z. M. Chen and J. Feng, An adaptive finite element algorithm with reliable and efficient error control for linear parabolic problems, Mathematics of Computation, 73 (2004), 1167-1193. doi: 10.1090/S0025-5718-04-01634-5. [8] A. Cohen, "Numerical Analysis of Wavelet Methods," Elsevier, 2003. [9] A. Cohen, W. Dahmen and R. DeVore, Adaptive wavelet methods for elliptic operator equations: Convergence rates, Mathematics of Computation, 70 (2001), 27-75. doi: 10.1090/S0025-5718-00-01252-7. [10] A. Cohen, W. Dahmen and R. DeVore, Adaptive wavelet methods II - Beyond the elliptic case, Foundations of Computational Mathematics, 2 (2002), 203-245. doi: 10.1007/s102080010027. [11] A. Cohen, I. Daubechies and J. C. Feauveau, Biorthogonal bases of compactly supported wavelets, Communications on Pure and Applied Mathematics, 45 (1992), 485-560. doi: 10.1002/cpa.3160450502. [12] A. Cohen, I. Daubechies and P. Vial, Wavelets on the interval and fast wavelet transforms, Applied and Computational Harmonic Analysis, 1 (1993), 54-81. doi: 10.1006/acha.1993.1005. [13] A. Cohen and R. Masson, Wavelet methods for second-order elliptic problems, preconditioning, and adaptivity, SIAM Journal on Scientific Computing, 21 (1999), 1006-1026. doi: 10.1137/S1064827597330613. [14] S. Dahlke, W. Dahmen, R. Hochmuth and R. Schneider, Stable multiscale bases and local error estimation for elliptic problems, Applied Numerical Mathematics, 23 (1997), 21-47. doi: 10.1016/S0168-9274(96)00060-8. [15] W. Dahmen and A. Kunoth, Adaptive wavelet methods for linear-quadratic elliptic control problems: convergence rates, SIAM Journal on Control and Optimization, 43 (2005), 1640-1675. doi: 10.1137/S0363012902419199. [16] W. Dahmen, A. Kunoth and K. Urban, Biorthogonal spline wavelets on the interval - Stability and moment conditions, Applied and Computational Harmonic Analysis, 6 (1999), 132-196. doi: 10.1006/acha.1998.0247. [17] W. Dahmen, S. Prossdorf and R. Schneider, Wavelet approximation methods for pseudo-differential eqautions II: Matrix compression and fast resolution, Advances in Computational Mathematics, 1 (1993), 259-335. doi: 10.1007/BF02072014. [18] I. Daubechies, Orthonormal bases of compactly supported wavelets, Communications on Pure and Applied Mathematics, 41 (1988), 909-996. doi: 10.1002/cpa.3160410705. [19] I. Daubechies, "Ten Lectures on Wavelets," SIAM Philadelphia, 1992. doi: 10.1137/1.9781611970104. [20] J. Liandrat and P. Tchamitchian, Resolution of the 1d regularized Burgers equation using a spatial wavelet approximation, Tech. Rep., NASA Contractor Report 187480, ICASE Report 90-83, NASA Langley Research Center, Hampton VA 23665-5225 (1990). [21] D. Liang, Q. Guo and S. Gong, A New Splitting Wavelet Method for Solving the General Aerosol Dynamics Equation, Journal of Aerosol Science, 39 (2008), 467-487. doi: 10.1016/j.jaerosci.2008.01.005. [22] P. Morin, R. H. Nochetto and K. G. Siebert, Data oscillation and convergence of adaptive FEM, SIAM Journal on Numerical Analysis, 38 (2000), 466-488. doi: 10.1137/S0036142999360044. [23] O. Roussel, K. Schneider, A. Tsigulin and H. Bockhorn, A conservative fully adaptive multiresolution algorithm for parabolic PDEs, Journal of Computational Physics, 188 (2003), 493-523.

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##### References:
 [1] J. M. Alam, N. K.-R. Kevlahan and O. V. Vasilyev, Simultaneous space-time adaptive wavelet solution of nonlinear parabolic differential equations, Journal of Computational Physics, 214 (2006), 829-857. doi: 10.1016/j.jcp.2005.10.009. [2] E. Bacry, S. Mallat and G. Papanicolaou, A wavelet based space-time adaptive numerical method for partial differential equation, Mathematical Modelling and Numerical Analysis, 26 (1992), 793-834. [3] A. Barinka, T. Barsch, P. Charton, A. Cohen, S. Dahlke, W. Dahmen and K. Urban, Adaptive wavelet schemes for elliptic problems-implementation and numerical experiments, SIAM Journal on Scientific Computing, 23 (2001), 910-939. doi: 10.1137/S1064827599365501. [4] A. Bindal, J. G. Khinast and M. G. Ierapetritou, Adaptive multiscale solution of dynamical systems in chemical processes using wavelets, Computers and Chemical Engineering, 27 (2003), 131-142. doi: 10.1016/S0098-1354(02)00165-5. [5] C. Canuto, A. Tabacco and K. Urban, The wavelet element method - Part I. Construction and analysis, Applied and Computational Harmonic Analysis, 6 (1999), 1-52. doi: 10.1006/acha.1997.0242. [6] J. M. Cascón, L. Ferragut and M. I. Asensio, Space-time adaptive algorithm for the mixed parabolic problem, Numerische Mathematik, 103 (2006), 367-392. doi: 10.1007/s00211-006-0677-y. [7] Z. M. Chen and J. Feng, An adaptive finite element algorithm with reliable and efficient error control for linear parabolic problems, Mathematics of Computation, 73 (2004), 1167-1193. doi: 10.1090/S0025-5718-04-01634-5. [8] A. Cohen, "Numerical Analysis of Wavelet Methods," Elsevier, 2003. [9] A. Cohen, W. Dahmen and R. DeVore, Adaptive wavelet methods for elliptic operator equations: Convergence rates, Mathematics of Computation, 70 (2001), 27-75. doi: 10.1090/S0025-5718-00-01252-7. [10] A. Cohen, W. Dahmen and R. DeVore, Adaptive wavelet methods II - Beyond the elliptic case, Foundations of Computational Mathematics, 2 (2002), 203-245. doi: 10.1007/s102080010027. [11] A. Cohen, I. Daubechies and J. C. Feauveau, Biorthogonal bases of compactly supported wavelets, Communications on Pure and Applied Mathematics, 45 (1992), 485-560. doi: 10.1002/cpa.3160450502. [12] A. Cohen, I. Daubechies and P. Vial, Wavelets on the interval and fast wavelet transforms, Applied and Computational Harmonic Analysis, 1 (1993), 54-81. doi: 10.1006/acha.1993.1005. [13] A. Cohen and R. Masson, Wavelet methods for second-order elliptic problems, preconditioning, and adaptivity, SIAM Journal on Scientific Computing, 21 (1999), 1006-1026. doi: 10.1137/S1064827597330613. [14] S. Dahlke, W. Dahmen, R. Hochmuth and R. Schneider, Stable multiscale bases and local error estimation for elliptic problems, Applied Numerical Mathematics, 23 (1997), 21-47. doi: 10.1016/S0168-9274(96)00060-8. [15] W. Dahmen and A. Kunoth, Adaptive wavelet methods for linear-quadratic elliptic control problems: convergence rates, SIAM Journal on Control and Optimization, 43 (2005), 1640-1675. doi: 10.1137/S0363012902419199. [16] W. Dahmen, A. Kunoth and K. Urban, Biorthogonal spline wavelets on the interval - Stability and moment conditions, Applied and Computational Harmonic Analysis, 6 (1999), 132-196. doi: 10.1006/acha.1998.0247. [17] W. Dahmen, S. Prossdorf and R. Schneider, Wavelet approximation methods for pseudo-differential eqautions II: Matrix compression and fast resolution, Advances in Computational Mathematics, 1 (1993), 259-335. doi: 10.1007/BF02072014. [18] I. Daubechies, Orthonormal bases of compactly supported wavelets, Communications on Pure and Applied Mathematics, 41 (1988), 909-996. doi: 10.1002/cpa.3160410705. [19] I. Daubechies, "Ten Lectures on Wavelets," SIAM Philadelphia, 1992. doi: 10.1137/1.9781611970104. [20] J. Liandrat and P. Tchamitchian, Resolution of the 1d regularized Burgers equation using a spatial wavelet approximation, Tech. Rep., NASA Contractor Report 187480, ICASE Report 90-83, NASA Langley Research Center, Hampton VA 23665-5225 (1990). [21] D. Liang, Q. Guo and S. Gong, A New Splitting Wavelet Method for Solving the General Aerosol Dynamics Equation, Journal of Aerosol Science, 39 (2008), 467-487. doi: 10.1016/j.jaerosci.2008.01.005. [22] P. Morin, R. H. Nochetto and K. G. Siebert, Data oscillation and convergence of adaptive FEM, SIAM Journal on Numerical Analysis, 38 (2000), 466-488. doi: 10.1137/S0036142999360044. [23] O. Roussel, K. Schneider, A. Tsigulin and H. Bockhorn, A conservative fully adaptive multiresolution algorithm for parabolic PDEs, Journal of Computational Physics, 188 (2003), 493-523.
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