Adaptive multiresolution finite element method based on second generation wavelets

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.