The objective of this paper is to develop a family of one-dimensional wavelet-based finite elements. First, independent wavelet bases are used to approximate displacement functions, unknown coefficients are determined through imposing the continuity, linear independence, completeness, and essential boundary conditions. A family of one-dimensional wavelet-based shape functions are then developed, which are hierarchical due to multiresolution property of wavelet. Secondly, to construct one-dimensional wavelet-based finite elements, derivation of the shape functions for a subdomain is employed. Thus, the wavelet-based finite elements being presented are embodied with properties in adaptivity as well as locality. Numerical examples are used to illustrate the characteristics of the current elements and to assess their accuracy and efficiency.
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