He, Yumin; Chen, Xuefeng; Xiang, Jiawei; He, Zhengjia Multiresolution analysis for finite element method using interpolating wavelet and lifting scheme. (English) Zbl 1156.65073 Commun. Numer. Methods Eng. 24, No. 11, 1045-1066 (2008). Summary: Based on the interpolating wavelet transform and a lifting scheme, a multiresolution analysis for finite element method is developed. By designing appropriate finite element interpolation functions using interpolating wavelet and lifted interpolating wavelet on the interval, the finite element equation may be scale decoupled via eliminating all coupling in the stiffness matrix of the element across scales, and then resolved in different spaces independently. The coarse solution can be obtained by solving the equation in the coarse approximation space, and refined by adding details, which can be obtained by solving the equations in the corresponding detail spaces, respectively. The method is well suited to the construction of adaptive algorithm and is powerful in analysing the field problems with changes in gradients and singularities. Numerical examples are given to verify the effectiveness of such a method. Cited in 1 Document MSC: 65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations 65L10 Numerical solution of boundary value problems involving ordinary differential equations 34B05 Linear boundary value problems for ordinary differential equations 65T60 Numerical methods for wavelets Keywords:lifting scheme; multiresolution analysis; finite element method; interpolating wavelet transform; numerical examples PDFBibTeX XMLCite \textit{Y. He} et al., Commun. Numer. Methods Eng. 24, No. 11, 1045--1066 (2008; Zbl 1156.65073) Full Text: DOI References: [1] Mallat, The theory for multiresolution signal decomposition: the wavelet representation, IEEE Transactions on Pattern Analysis and Machine Intelligence 11 (7) pp 674– (1989) · Zbl 0709.94650 [2] Ko, A class of finite element methods based on orthonormal, compactly supported wavelets, Computational Mechanics 16 pp 235– (1995) · Zbl 0830.65084 [3] Chen, The construction of wavelet finite element and its application, Finite Elements in Analysis and Design 40 pp 541– (2004) [4] Han, A spline wavelet finite-element method in structural mechanics, International Journal for Numerical Methods in Engineering 66 pp 166– (2006) · Zbl 1110.74851 [5] Sweldens, The lifting scheme: a construction of second generation wavelets, SIAM Journal on Mathematical Analysis 29 (2) pp 511– (1998) · Zbl 0911.42016 [6] Sweldens, The lifting scheme: a custom-design construction of biorthogonal wavelets, Applied and Computational Harmonic Analysis 3 (2) pp 186– (1996) · Zbl 0874.65104 [7] Sweldens, ACM SIGGRAPH Course Notes, in: Wavelets in Computer Graphics pp 15– (1996) [8] D’Heedene, Generalized hierarchical bases: a Wavelet-Ritz-Galerkin framework for Lagrangian FEM, Engineering Computations: International Journal for Computer-aided Engineering and Software 22 (1) pp 15– (2005) [9] Amaratunga, Multiresolution modeling with operator-customized wavelets derived from finite elements, Computer Methods in Applied Mechanics and Engineering 195 pp 2509– (2006) · Zbl 1125.74051 [10] Sudarshan, A combined approach for goal-oriented error estimation and adaptivity using operator-customized finite element wavelets, International Journal for Numerical Methods in Engineering 66 pp 1002– (2006) · Zbl 1110.74857 [11] Mallat, A Wavelet Tour of Signal Processing (2003) [12] Donoho, Technical Report 408 (1992) [13] Harten, Adaptive multiresolution schemes for shock computations, Journal of Computational Physics 115 pp 319– (1994) · Zbl 0925.65151 [14] Deslauriers, Fractals, Dimensions Non Entières et Applications pp 44– (1987) [15] Deslauriers, Symmetric iterative interpolation processes, Constructive Approximation 5 (1) pp 49– (1989) · Zbl 0659.65004 [16] Carnicer, Local decompositions of refinable spaces, Applied and Computational Harmonic Analysis 3 pp 125– (1996) · Zbl 0859.42025 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.