The use of wavelet-like basis functions in finite element algorithms

In this research, wavelet-like functions are used as the basis functions in the finite element solution of partial differential equations. As computing power continues to increase, researchers are investigating problems with much greater complexity, and large discretizations containing millions of unknowns are possible. The matrix equation resulting from the finite element method can become ill-conditioned as the number of unknowns increases. If iterative techniques are used to solve the matrix equation, many iterations of the technique can be required before an accurate solution is obtained. If the matrix is significantly ill-conditioned, lack of convergence is even possible. Using the wavelet-like basis can alleviate the problems associated with stability and convergence. In this research, several examples illustrating the beneficial effects of the wavelet-like basis are considered. As the number of wavelet-like basis functions is increased, the condition number of the preconditioned matrix and the number of steps required for the conjugate gradient method to converge are observed. Comparisons with analytic solutions and numerical solutions obtained with the traditional basis functions are made.; In addition, the generation and use of higher order wavelet-like functions is discussed. These higher order functions result in increased accuracy in the numerical solutions. To illustrate the increased accuracy, the solutions obtained using higher order wavelet-like functions are compared to solutions obtained using linear basis functions. Comparisons of the condition number of the preconditioned matrix and the number of steps required for convergence are also presented.; Several advantages and disadvantages of the wavelet-like basis are discussed. One of these disadvantages is the time required to generate the basis. This is especially true for two-dimensional (or higher) basis generation. An alternate technique for generating higher dimensional wavelet-like functions is proposed, and results are discussed. Comparisons of the condition number of the preconditioned matrix and the number of steps required for convergence for the two-dimensional basis functions generated using the new method are presented.; Finally, a finite element time domain (FETD) technique utilizing the wavelet-like basis functions is formulated. An investigation of the allowable time step size when the cubic wavelet-like functions are used in the algorithm is made. Comparisons between solutions obtained with the FETID algorithm utilizing cubic wavelet-like functions, analytic solutions, and solutions obtained by the traditional finite difference time domain (FDTD) method are presented.