Date of Award:

12-2018

Document Type:

Dissertation

Degree Name:

Doctor of Philosophy (PhD)

Department:

Mathematics and Statistics

Advisor/Chair:

Joseph V. Koebbe

Abstract

This dissertation considers an approximation strategy using a wavelet reconstruction scheme for solving elliptic problems. The foci of the work are on (1) the approximate solution of differential equations using multiresolution analysis based on wavelet transforms and (2) the homogenization process for solving one and two-dimensional problems, to understand the solutions of second order elliptic problems. We employed homogenization to compute the average formula for permeability in a porous medium. The structure of the associated multiresolution analysis allows for the reconstruction of the approximate solution of the primary variable in the elliptic equation. Using a one-dimensional wavelet reconstruction algorithm proposed in this work, we are able to numerically compute the approximations of the pressure variables. This algorithm can directly be applied to elliptic problems with discontinuous coefficients.We also implemented Java codes to solve the two dimensional elliptic problems using our methods of solutions. Furthermore, we propose homogenization wavelet reconstruction algorithm, fast transform and the inverse transform algorithms that use the results from the solutions of the local problems and the partial derivatives of the pressure variables to reconstruct the solutions.

Checksum

d915421f54adc4f1455dab374b775663

Included in

Mathematics Commons

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