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A Fast Wavelet-multigrid Method To Solve Elliptic Partial Differential Equations

N. Bujurke, C. Salimath, Ramesh B. Kudenatti, S. Shiralashetti
Published 2007 · Mathematics, Computer Science

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Abstract In this paper, we present a wavelet-based multigrid approach to solve elliptic boundary value problems encountered in mathematical physics. The system of equations arising from finite difference discretization is represented in wavelet-basis. These equations are solved using multiresolution properties of wavelets characterized by sparse matrices having condition number O(1) together with a multigrid strategy for accelerating convergence. The filter coefficients of D 2 k , k  = 2, 3, 4 from Daubechies family of wavelets are used to demonstrate the effectiveness and efficiency of the method. The distinguishing feature of the method is; it works as both solver and preconditioner. As a consequence, it avoids instability, minimizes error and speeds up convergence. Compared to the classical multigrid method, this approach requires substantially shorter computation time; at the same time meeting accuracy requirements. It is found that just one cycle is enough for the convergence of wavelet-multigrid scheme whereas normally 7–8 cycles are required in classical multigrid schemes to meet the same accuracy. Numerical examples show that, the scheme offers a fast and robust technique for elliptic pde’s.
This paper references
10.1002/CPA.3160410705
Orthonormal bases of compactly supported wavelets
I. Daubechies (1988)
10.1109/34.192463
A Theory for Multiresolution Signal Decomposition: The Wavelet Representation
S. Mallat (1989)
10.1007/BF01409786
The frequency decomposition multi-grid method
W. Hackbusch (1989)
An Introduction to Multigrid Methods
P. Wesseling (1992)
10.1007/BF01385864
Multilevel preconditioning
W. Dahmen (1992)
10.1137/0729059
Wavelet methods for fast resolution of elliptic problems
S. Jaffard (1992)
10.1007/BF01385851
Galerkin-wavelet methods for two-point boundary value problems
J. Xu (1992)
10.1137/0914031
Wavelets and Multigrid
W. Briggs (1993)
10.1006/JCPH.1995.1147
A multilevel wavelet collocation method for solving partial differential equations in a finite domain
Oleg V. Vasilyev (1995)
10.1109/83.388081
Multiresolution tomographic reconstruction using wavelets
A. H. Delaney (1995)
10.1006/JCPH.1996.0111
A Dynamically Adaptive Multilevel Wavelet Collocation Method for Solving Partial Differential Equations in a Finite Domain
Oleg V. Vasilyev (1996)
10.1109/42.563666
A wavelet-based multiresolution regularized least squares reconstruction approach for optical tomography
Wenwu Zhu (1997)
Wavelet transforms - introduction to theory and applications
R. Rao (1997)
10.1063/1.168739
Solution of multiscale partial differential equations using wavelets
S. Goedecker (1998)
10.1002/NME.352
Hat interpolation wavelet‐based multi‐scale Galerkin method for thin‐walled box beam analysis
Y. Kim (2002)
10.1142/S021963360300063X
Combining multigrid and wavelet ideas to construct more efficient multiscale algorithms
S. Goedecker (2002)
10.1016/B978-008044046-0.50524-8
A multiresolution finite element method using second generation Hermite multiwavelets
R. Sudarshan (2003)
10.1016/S0096-3003(02)00845-7
Wavelet based multigrid methods for linear and nonlinear elliptic partial differential equations
A. Avudainayagam (2004)
10.1016/j.amc.2005.11.048
An analysis of rough poroelastic bearings with reference to lubrication mechanism of synovial joints
N. Bujurke (2006)
10.1016/j.amc.2005.06.007
Surface roughness effects on squeeze film poroelastic bearings
N. Bujurke (2006)



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10.1007/s40819-020-00879-2
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10.1063/5.0014591
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10.26637/mjm0s20/0033
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10.1142/S2047684118500306
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10.1142/S1793557119500542
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10.1142/S0219691318500467
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S. C. Shiralashetti (2018)
10.22124/JMM.2018.5019.1059
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10.1016/J.AEJ.2016.12.007
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S. C. Shiralashetti (2017)
10.18052/WWW.SCIPRESS.COM/BMSA.18.50
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S. Shiralashetti (2017)
10.1080/02522667.2016.1190568
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S. Shiralashetti (2017)
Daubechies wavelet based full approximation scheme for solving Burgers’ equation arising in Fluid Dynamics
S. C. Shiralashetti (2017)
LEIBNITZ-HAAR WAVELET COLLOCATION METHOD FOR THE NUMERICAL SOLUTION OF NONLINEAR FREDHOLM INTEGRAL EQUATIONS
S. C. Shiralashetti (2016)
10.3390/LUBRICANTS4010009
Investigation of Couple Stress Fluid and Surface Roughness Effects in the Elastohydrodynamic Lubrication Problems using Wavelet-Based Decoupled Method
S. Shiralashetti (2016)
10.7726/AJHMT.2016.1018
Wavelet Based Numerical Solution of Elasto- hydrodynamic Lubrication Problems via Lifting Scheme
S. C. Shiralashetti (2016)
10.7726/AJHMT.2016.1010
A New Daubechies Orthogonal Discrete Wavelet Transform with Permutation Preconditioner Method for the Numerical Solution of EHL Problems
S. C. Shiralashetti (2016)
10.1016/J.ASEJ.2016.04.024
Wavelet based decoupled method for the investigation of surface roughness effects in elastohydrodynamic lubrication problems using couple stress fluid
S. C. Shiralashetti (2016)
Solution of Non-Homogeneous Burgers Equation by Haar Wavelet Method
Sumana R. Shesha (2016)
10.21042/AMNS.2016.2.00042
Modified Wavelet Full-Approximation Scheme for the Numerical Solution of Nonlinear Volterra integral and integro-differential Equations
S. Shiralashetti (2016)
10.1016/J.AEJ.2016.07.019
New wavelet based full-approximation scheme for the numerical solution of nonlinear elliptic partial differential equations
S. C. Shiralashetti (2016)
10.1016/J.JQSRT.2013.09.003
Application of the multigrid method in a deterministic solution scheme for the three-dimensional radiative transfer equation
Haruma Ishida (2014)
10.1007/978-3-319-05657-9_13
Wavelet-Multigrid Method for Solving Modified Reynolds Equation Modeling Synovial Fluid Flow in a Normal Human Knee Joint
Chandrasekhar Salimath (2014)
10.22080/CJMS.2017.1668
Application of Daubechies wavelets for solving Kuramoto-Sivashinsky type equations
Ali Davari (2014)
An application of the Daubechies Orthogonal Wavelets in Power system Engineering
S. Shiralashetti (2014)
Wavelets : Algorithms and Applications in Science and Technology
C. Salimath (2014)
Solving PDEs with the Aid of Two-Dimensional
Haar Wavelets (2014)
10.1016/J.JMAA.2011.10.057
A fast wavelet block Jacobi method
Dongsheng Cheng (2012)
10.1007/S10409-012-0045-3
A wavelet approach for active-passive vibration control of laminated plates
Jizeng Wang (2012)
FAST WAVELET MULTIGRID SOLUTION OF THE MODIFIED REYNOLDS EQUATION OF MAGNETOHYDRODYNAMIC LUBRICATION FLOW BETWEEN ROUGH PLATES
S. C. Shiralashetti (2012)
10.1016/j.amc.2012.08.027
Cubic spline wavelets with complementary boundary conditions
D. Cerná (2012)
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