# 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
##### 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.  doi: 10.1016/j.jcp.2005.10.009.  Google Scholar [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.   Google Scholar [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.  doi: 10.1137/S1064827599365501.  Google Scholar [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.  doi: 10.1016/S0098-1354(02)00165-5.  Google Scholar [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.  doi: 10.1006/acha.1997.0242.  Google Scholar [6] J. M. Cascón, L. Ferragut and M. I. Asensio, Space-time adaptive algorithm for the mixed parabolic problem,, Numerische Mathematik, 103 (2006), 367.  doi: 10.1007/s00211-006-0677-y.  Google Scholar [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.  doi: 10.1090/S0025-5718-04-01634-5.  Google Scholar [8] A. Cohen, "Numerical Analysis of Wavelet Methods,", Elsevier, (2003).   Google Scholar [9] A. Cohen, W. Dahmen and R. DeVore, Adaptive wavelet methods for elliptic operator equations: Convergence rates,, Mathematics of Computation, 70 (2001), 27.  doi: 10.1090/S0025-5718-00-01252-7.  Google Scholar [10] A. Cohen, W. Dahmen and R. DeVore, Adaptive wavelet methods II - Beyond the elliptic case,, Foundations of Computational Mathematics, 2 (2002), 203.  doi: 10.1007/s102080010027.  Google Scholar [11] A. Cohen, I. Daubechies and J. C. Feauveau, Biorthogonal bases of compactly supported wavelets,, Communications on Pure and Applied Mathematics, 45 (1992), 485.  doi: 10.1002/cpa.3160450502.  Google Scholar [12] A. Cohen, I. Daubechies and P. Vial, Wavelets on the interval and fast wavelet transforms,, Applied and Computational Harmonic Analysis, 1 (1993), 54.  doi: 10.1006/acha.1993.1005.  Google Scholar [13] A. Cohen and R. Masson, Wavelet methods for second-order elliptic problems, preconditioning, and adaptivity,, SIAM Journal on Scientific Computing, 21 (1999), 1006.  doi: 10.1137/S1064827597330613.  Google Scholar [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.  doi: 10.1016/S0168-9274(96)00060-8.  Google Scholar [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.  doi: 10.1137/S0363012902419199.  Google Scholar [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.  doi: 10.1006/acha.1998.0247.  Google Scholar [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.  doi: 10.1007/BF02072014.  Google Scholar [18] I. Daubechies, Orthonormal bases of compactly supported wavelets,, Communications on Pure and Applied Mathematics, 41 (1988), 909.  doi: 10.1002/cpa.3160410705.  Google Scholar [19] I. Daubechies, "Ten Lectures on Wavelets,", SIAM Philadelphia, (1992).  doi: 10.1137/1.9781611970104.  Google Scholar [20] J. Liandrat and P. Tchamitchian, Resolution of the 1d regularized Burgers equation using a spatial wavelet approximation,, Tech. Rep., (1990), 90.   Google Scholar [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.  doi: 10.1016/j.jaerosci.2008.01.005.  Google Scholar [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.  doi: 10.1137/S0036142999360044.  Google Scholar [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.   Google Scholar

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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.  doi: 10.1016/j.jcp.2005.10.009.  Google Scholar [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.   Google Scholar [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.  doi: 10.1137/S1064827599365501.  Google Scholar [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.  doi: 10.1016/S0098-1354(02)00165-5.  Google Scholar [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.  doi: 10.1006/acha.1997.0242.  Google Scholar [6] J. M. Cascón, L. Ferragut and M. I. Asensio, Space-time adaptive algorithm for the mixed parabolic problem,, Numerische Mathematik, 103 (2006), 367.  doi: 10.1007/s00211-006-0677-y.  Google Scholar [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.  doi: 10.1090/S0025-5718-04-01634-5.  Google Scholar [8] A. Cohen, "Numerical Analysis of Wavelet Methods,", Elsevier, (2003).   Google Scholar [9] A. Cohen, W. Dahmen and R. DeVore, Adaptive wavelet methods for elliptic operator equations: Convergence rates,, Mathematics of Computation, 70 (2001), 27.  doi: 10.1090/S0025-5718-00-01252-7.  Google Scholar [10] A. Cohen, W. Dahmen and R. DeVore, Adaptive wavelet methods II - Beyond the elliptic case,, Foundations of Computational Mathematics, 2 (2002), 203.  doi: 10.1007/s102080010027.  Google Scholar [11] A. Cohen, I. Daubechies and J. C. Feauveau, Biorthogonal bases of compactly supported wavelets,, Communications on Pure and Applied Mathematics, 45 (1992), 485.  doi: 10.1002/cpa.3160450502.  Google Scholar [12] A. Cohen, I. Daubechies and P. Vial, Wavelets on the interval and fast wavelet transforms,, Applied and Computational Harmonic Analysis, 1 (1993), 54.  doi: 10.1006/acha.1993.1005.  Google Scholar [13] A. Cohen and R. Masson, Wavelet methods for second-order elliptic problems, preconditioning, and adaptivity,, SIAM Journal on Scientific Computing, 21 (1999), 1006.  doi: 10.1137/S1064827597330613.  Google Scholar [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.  doi: 10.1016/S0168-9274(96)00060-8.  Google Scholar [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.  doi: 10.1137/S0363012902419199.  Google Scholar [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.  doi: 10.1006/acha.1998.0247.  Google Scholar [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.  doi: 10.1007/BF02072014.  Google Scholar [18] I. Daubechies, Orthonormal bases of compactly supported wavelets,, Communications on Pure and Applied Mathematics, 41 (1988), 909.  doi: 10.1002/cpa.3160410705.  Google Scholar [19] I. Daubechies, "Ten Lectures on Wavelets,", SIAM Philadelphia, (1992).  doi: 10.1137/1.9781611970104.  Google Scholar [20] J. Liandrat and P. Tchamitchian, Resolution of the 1d regularized Burgers equation using a spatial wavelet approximation,, Tech. Rep., (1990), 90.   Google Scholar [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.  doi: 10.1016/j.jaerosci.2008.01.005.  Google Scholar [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.  doi: 10.1137/S0036142999360044.  Google Scholar [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.   Google Scholar
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