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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 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. 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. 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. 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. 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. 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. 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. 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. doi: 10.1007/BF02072014.

[18]

I. Daubechies, Orthonormal bases of compactly supported wavelets,, Communications on Pure and Applied Mathematics, 41 (1988), 909. 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., (1990), 90.

[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.

[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.

[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.

show all references

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 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. 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. 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. 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. 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. 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. 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. 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. doi: 10.1007/BF02072014.

[18]

I. Daubechies, Orthonormal bases of compactly supported wavelets,, Communications on Pure and Applied Mathematics, 41 (1988), 909. 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., (1990), 90.

[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.

[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.

[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.

[1]

Jong-Shenq Guo, Satoshi Sasayama, Chi-Jen Wang. Blowup rate estimate for a system of semilinear parabolic equations. Communications on Pure & Applied Analysis, 2009, 8 (2) : 711-718. doi: 10.3934/cpaa.2009.8.711

[2]

Gary Lieberman. A new regularity estimate for solutions of singular parabolic equations. Conference Publications, 2005, 2005 (Special) : 605-610. doi: 10.3934/proc.2005.2005.605

[3]

Shuai Ren, Tao Zhang, Fangxia Shi. Characteristic analysis of carrier based on the filtering and a multi-wavelet method for the information hiding. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1291-1299. doi: 10.3934/dcdss.2015.8.1291

[4]

Ugur G. Abdulla. On the optimal control of the free boundary problems for the second order parabolic equations. II. Convergence of the method of finite differences. Inverse Problems & Imaging, 2016, 10 (4) : 869-898. doi: 10.3934/ipi.2016025

[5]

Wen-ming He, Jun-zhi Cui. The estimate of the multi-scale homogenization method for Green's function on Sobolev space $W^{1,q}(\Omega)$. Communications on Pure & Applied Analysis, 2012, 11 (2) : 501-516. doi: 10.3934/cpaa.2012.11.501

[6]

Ugur G. Abdulla. On the optimal control of the free boundary problems for the second order parabolic equations. I. Well-posedness and convergence of the method of lines. Inverse Problems & Imaging, 2013, 7 (2) : 307-340. doi: 10.3934/ipi.2013.7.307

[7]

Patrick Henning, Mario Ohlberger. A-posteriori error estimate for a heterogeneous multiscale approximation of advection-diffusion problems with large expected drift. Discrete & Continuous Dynamical Systems - S, 2016, 9 (5) : 1393-1420. doi: 10.3934/dcdss.2016056

[8]

Xingwen Hao, Yachun Li, Qin Wang. A kinetic approach to error estimate for nonautonomous anisotropic degenerate parabolic-hyperbolic equations. Kinetic & Related Models, 2014, 7 (3) : 477-492. doi: 10.3934/krm.2014.7.477

[9]

José A. Carrillo, Jean Dolbeault, Ivan Gentil, Ansgar Jüngel. Entropy-energy inequalities and improved convergence rates for nonlinear parabolic equations. Discrete & Continuous Dynamical Systems - B, 2006, 6 (5) : 1027-1050. doi: 10.3934/dcdsb.2006.6.1027

[10]

Niklas Hartung. Efficient resolution of metastatic tumor growth models by reformulation into integral equations. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 445-467. doi: 10.3934/dcdsb.2015.20.445

[11]

Stanisław Migórski, Shengda Zeng. The Rothe method for multi-term time fractional integral diffusion equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 719-735. doi: 10.3934/dcdsb.2018204

[12]

Guillaume Bal, Jiaming Chen, Anthony B. Davis. Reconstruction of cloud geometry from high-resolution multi-angle images. Inverse Problems & Imaging, 2018, 12 (2) : 261-280. doi: 10.3934/ipi.2018011

[13]

Thierry Cazenave, Flávio Dickstein, Fred B. Weissler. Multi-scale multi-profile global solutions of parabolic equations in $\mathbb{R}^N $. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 449-472. doi: 10.3934/dcdss.2012.5.449

[14]

Stephan Didas, Joachim Weickert. Integrodifferential equations for continuous multiscale wavelet shrinkage. Inverse Problems & Imaging, 2007, 1 (1) : 47-62. doi: 10.3934/ipi.2007.1.47

[15]

Stefano Galatolo, Isaia Nisoli, Benoît Saussol. An elementary way to rigorously estimate convergence to equilibrium and escape rates. Journal of Computational Dynamics, 2015, 2 (1) : 51-64. doi: 10.3934/jcd.2015.2.51

[16]

Tohru Nakamura, Shinya Nishibata. Energy estimate for a linear symmetric hyperbolic-parabolic system in half line. Kinetic & Related Models, 2013, 6 (4) : 883-892. doi: 10.3934/krm.2013.6.883

[17]

Sun-Sig Byun, Yunsoo Jang. Calderón-Zygmund estimate for homogenization of parabolic systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 6689-6714. doi: 10.3934/dcds.2016091

[18]

Li-Ming Yeh. Pointwise estimate for elliptic equations in periodic perforated domains. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1961-1986. doi: 10.3934/cpaa.2015.14.1961

[19]

Liangjun Weng. The interior gradient estimate for some nonlinear curvature equations. Communications on Pure & Applied Analysis, 2019, 18 (4) : 1601-1612. doi: 10.3934/cpaa.2019076

[20]

Limei Dai. Multi-valued solutions to a class of parabolic Monge-Ampère equations. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1061-1074. doi: 10.3934/cpaa.2014.13.1061

 Impact Factor: 

Metrics

  • PDF downloads (5)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]