Wavelet-finite element bases for numerical solutions of partial differential equations

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Strader, Roy Arlen
Texas A&M University
mathematics.; Major mathematics.
thesis / dissertation description
This thesis presents a method to solve elliptic partial differential equations (PDES) using wavelets and finite elements. We focus on third order Daubechies' wavelets and piecewise linear finite elements. Functions from these two bases will be chosen carefully and will be combined into one basis. This hybrid basis is used in the Ritz-Galerkin method for numerically solving PDES. Three different procedures for choosing which functions to use from each basis at each level of refinement will be discussed and numerical results will be used to illustrate the strengths and weaknesses of each variation. We show which of these strategies will be the most effective and prove that with appropriate smoothness requirements the convergence rate for this best choice matches that of standard Finite Element Methods (FEM) with piecewise linear elements. Finally, we compare other computational measures such as the conditioning of the Galerkin matrices and the overall complexity of the algorithms with the same measures for FEM.