On the multi-level splitting of finite element spaces
Numerische Mathematik
A Theory for Multiresolution Signal Decomposition: The Wavelet Representation
IEEE Transactions on Pattern Analysis and Machine Intelligence
ICIAM 91 Proceedings of the second international conference on Industrial and applied mathematics
The lifting scheme: a construction of second generation wavelets
SIAM Journal on Mathematical Analysis
Adaptive Wavelet Schemes for Elliptic Problems---Implementation and Numerical Experiments
SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
A study of the construction and application of a Daubechies wavelet-based beam element
Finite Elements in Analysis and Design
The construction of wavelet finite element and its application
Finite Elements in Analysis and Design
An adaptive multilevel wavelet collocation method for elliptic problems
Journal of Computational Physics
A Wavelet Tour of Signal Processing, Third Edition: The Sparse Way
A Wavelet Tour of Signal Processing, Third Edition: The Sparse Way
Finite Elements in Analysis and Design
A second generation wavelet based finite elements on triangulations
Computational Mechanics
Review: Wavelet-based numerical analysis: A review and classification
Finite Elements in Analysis and Design
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A distinguishing feature of second generation wavelets is that it can be custom designed depending on applications. Based on second generation wavelets, a multiresolution finite element method is discussed, and its adaptive algorithm is constructed. The hierarchical approximation spaces for finite element analysis are produced. The finite element equation is scale-decoupled via eliminating all coupling in the stiffness matrix of element across scales, then resolved in different spaces independently. The coarse solution can be obtained in the coarse approximation space, and refined by adding details in the detail spaces over several levels till the equation is resolved to the desired accuracy. The scale-decoupling condition of the stiffness matrix of element is proposed by introducing wavelet vanishing moments, and the principle of constructing the scale-decoupling wavelet bases is established. The method establishes an important connection between finite element analysis and multiresolution analysis. The numerical examples have illustrated that the proposed method is powerful to analyze the field problems with changes in gradients and singularities.