Affine Fractal Interpolation Functions and Wavelet-Based Finite Elements

摘要

This paper derives a class of finite elements from wavelets generated using affine, fractal interpolation functions (AFIF). The finite elements derived from wavelets generated from an AFIF differ from recent elements derived by the authors in that multivalued scaling functions are employed. These elements are similar to conventional finite elements in that they are compactly supported, and are interpolatory at dyadic points which define the nodes. Specifically, two scaling functions and two wavelets that are Lipschitz continuous are employed. Each scaling function has support of length one or two, and is either symmetric or antisymmetric. These properties enables the incorporation of both Dirichlet or Neumann boundary conditions in a straightforward manner. For either class of boundary conditions, the multiresolution V_k+1 ident to K_k direct + W_k restricted to the compact interval of interest is fully orthogonal. This fact is advantageous in that it facilitates both the analysis of the convergence of the multigrid methods, as well as enabling an efficient implementation.