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In the computational sciences, both error and wavelet analysis have received abundant attention in the scientific literature. Wavelets have been applied in a wide range of areas such as time-domain analysis, signal compression, and the numerical solution of partial differential equations and integral equations.For instance, wavelet-like basis functions have been used in the numerical solution of differential equations and the error introduced by them has been investigated. Error analysis and Richardson extrapolation have also been used to reduce the numerical error due to the use of first order wavelet-like basis functions. In the present paper, the same techniques will be applied to reduce the numerical error arising when higher order wavelet-Uike basis functions are used. The numerical error introduced by the higher order wavelet-like basis functions will be discussed. The formation of the Richardson extrapolate, which is found from solutions obtained a t different levels of the wavelet analysis, will also be investigated. Finally, a discussion of the error of the Richardson extrapolate will be presented. Example problems will be considered to illustrate these ideas. I. GENERATION OF BASIS FUNCTIONSThe generalization of generating higher order wavelet-like functions proceeds in much the same way as discussed in [l]. The generalization of the multiresolution analysis ( M U ) is described. In the beginning, a coarse discretization of the problem domain is defined. The process begins by taking the traditional higher order basis functions associated with this mesh. These functions are then orthonormalized against each other. The orthonormal functions are considered the first higher order wavelet-like basis. Now, each segment in the original mesh is divided into two equal segments. The nodes that are associated with each of these segments are examined. Some of the nodes in the segments will have been nodes in the previous discretization, while other nodes will not have been associated with the previous 0-7803-7339-1/02/$17.00 02002 IEEE segments.Only those nodes that were not previously in the mesh are considered.The traditional' higher order basis functions associated with these added nodes are then orthogonalized against all of the previously found wavelet-like basis functions. Finally, the resulting functions are orthonormalized against each other.These functions are then added to the wavelet-like basis. This concludes the first added level in the MRA. The procedure of dividing each segment, orthogonalizing the associated traditional basis functions, and then orthonormalizing the resulting functions can be repeated as necessary until the desired discretization of the problem domain is obtained.The procedure described above can best be illustrated through an example. Consider the original discretization of the problem domain to be the single segment0 S x 5 3 and that cubic wavelet-like functions are to be generated. Dirichlet boundary conditions will be considered at the endpoints of the domain. Therefore, the nod...
In the computational sciences, both error and wavelet analysis have received abundant attention in the scientific literature. Wavelets have been applied in a wide range of areas such as time-domain analysis, signal compression, and the numerical solution of partial differential equations and integral equations.For instance, wavelet-like basis functions have been used in the numerical solution of differential equations and the error introduced by them has been investigated. Error analysis and Richardson extrapolation have also been used to reduce the numerical error due to the use of first order wavelet-like basis functions. In the present paper, the same techniques will be applied to reduce the numerical error arising when higher order wavelet-Uike basis functions are used. The numerical error introduced by the higher order wavelet-like basis functions will be discussed. The formation of the Richardson extrapolate, which is found from solutions obtained a t different levels of the wavelet analysis, will also be investigated. Finally, a discussion of the error of the Richardson extrapolate will be presented. Example problems will be considered to illustrate these ideas. I. GENERATION OF BASIS FUNCTIONSThe generalization of generating higher order wavelet-like functions proceeds in much the same way as discussed in [l]. The generalization of the multiresolution analysis ( M U ) is described. In the beginning, a coarse discretization of the problem domain is defined. The process begins by taking the traditional higher order basis functions associated with this mesh. These functions are then orthonormalized against each other. The orthonormal functions are considered the first higher order wavelet-like basis. Now, each segment in the original mesh is divided into two equal segments. The nodes that are associated with each of these segments are examined. Some of the nodes in the segments will have been nodes in the previous discretization, while other nodes will not have been associated with the previous 0-7803-7339-1/02/$17.00 02002 IEEE segments.Only those nodes that were not previously in the mesh are considered.The traditional' higher order basis functions associated with these added nodes are then orthogonalized against all of the previously found wavelet-like basis functions. Finally, the resulting functions are orthonormalized against each other.These functions are then added to the wavelet-like basis. This concludes the first added level in the MRA. The procedure of dividing each segment, orthogonalizing the associated traditional basis functions, and then orthonormalizing the resulting functions can be repeated as necessary until the desired discretization of the problem domain is obtained.The procedure described above can best be illustrated through an example. Consider the original discretization of the problem domain to be the single segment0 S x 5 3 and that cubic wavelet-like functions are to be generated. Dirichlet boundary conditions will be considered at the endpoints of the domain. Therefore, the nod...
I n this paper, some of the advantages and disadvantages of the use of wavelet-like basis functions are discussed. Two modifications for mitigating some of the disadvantages are considered. Numerical results obtained using these modifications are presented.
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