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
Wavelets for computer graphics: theory and applications
Wavelets for computer graphics: theory and applications
The lifting scheme: a construction of second generation wavelets
SIAM Journal on Mathematical Analysis
Second-generation wavelet collocation method for the solution of partial differential equations
Journal of Computational Physics
Linear Independence and Stability of Piecewise Linear Prewavelets on Arbitrary Triangulations
SIAM Journal on Numerical Analysis
Towards a Realization of a Wavelet Galerkin Method on Non-Trivial Domains
Journal of Scientific Computing
The construction of wavelet finite element and its application
Finite Elements in Analysis and Design
C1 hierarchical Riesz bases of Lagrange type on Powell-Sabin triangulations
Journal of Computational and Applied Mathematics
Adaptive multiresolution finite element method based on second generation wavelets
Finite Elements in Analysis and Design
A new wavelet multigrid method
Journal of Computational and Applied Mathematics
A modified wavelet approximation of deflections for solving PDEs of beams and square thin plates
Finite Elements in Analysis and Design
Daubechies wavelet beam and plate finite elements
Finite Elements in Analysis and Design
Finite element wavelets with improved quantitative properties
Journal of Computational and Applied Mathematics
A multivariable wavelet-based finite element method and its application to thick plates
Finite Elements in Analysis and Design
Review: Wavelet-based numerical analysis: A review and classification
Finite Elements in Analysis and Design
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In this paper we have developed a second generation wavelet based finite element method for solving elliptic PDEs on two dimensional triangulations using customized operator dependent wavelets. The wavelets derived from a Courant element are tailored in the second generation framework to decouple some elliptic PDE operators. Starting from a primitive hierarchical basis the wavelets are lifted (enhanced) to achieve local scale-orthogonality with respect to the operator of the PDE. The lifted wavelets are used in a Galerkin type discretization of the PDE which result in a block diagonal, sparse multiscale stiffness matrix. The blocks corresponding to different resolutions are completely decoupled, which makes the implementation of new wavelet finite element very simple and efficient. The solution is enriched adaptively and incrementally using finer scale wavelets. The new procedure completely eliminates wastage of resources associated with classical finite element refinement. Finally some numerical experiments are conducted to analyze the performance of this method.