In this paper we combine biorthogonal wavelet systems with the philosophy of Spectral Element Method to obtain a biorthogonal wavelet system on fairly general bounded domains. We also extend the boundary adaption of wavelet elements to first order derivatives allowing the construction of basis functions that exactly satisfy boundary conditions. Since this method allows us to take advantage of structural features of phononic crystals and the boundary conditions are satisfied rigorously, a better accuracy and higher efficiency can be obtained.

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