289 research outputs found

    Criteria for Hierarchical Bases in Sobolev Spaces

    Get PDF
    AbstractSeveral approaches to solving elliptic problems numerically are based on hierarchical Riesz bases in Sobolev spaces. We are interested in determining the exact range of Sobolev exponents for which a system of compactly supported functions derived from a multiresolution analysis forms such a Riesz basis. This involves determining the smoothness of the dual system. The elements of the dual system typically consist of noncompactly supported functions, whose smoothness can be treated by extending the results of 7, 9, and 22. We show how to determine the exact range of Sobolev exponents in the multivariate case, both theoretically and numerically, from spectral properties of transfer operators. This technique is applied to several bases deriving from linear finite elements which have been proposed in the literature. For 29hierarchical basis, we find that it forms a Riesz basis in Hs(Rd) for −0.990236…<s<3/2

    Fast numerical methods for non-local operators

    Full text link

    Fast Numerical Methods for Non-local Operators

    Get PDF
    [no abstract available

    Post-Processing Techniques and Wavelet Applications for Hammerstein Integral Equations

    Get PDF
    This dissertation is focused on the varieties of numerical solutions of nonlinear Hammerstein integral equations. In the first part of this dissertation, several acceleration techniques for post-processed solutions of the Hammerstein equation are discussed. The post-processing techniques are implemented based on interpolation and extrapolation. In this connection, we generalize the results in [29] and [28] to nonlinear integral equations of the Hammerstein type. Post-processed collocation solutions are shown to exhibit better accuracy. Moreover, an extrapolation technique for the Galerkin solution of Hammerstein equation is also obtained. This result appears new even in the setting of the linear Fredholm equation. In the second half of this dissertation, the wavelet-collocation technique of solving nonlinear Hammerstein integral equation is discussed. The main objective is to establish a fast wavelet-collocation method for Hammerstein equation by using a \u27linearization\u27 technique. The sparsity in the Jacobian matrix takes place in the fast wavelet-collocation method for Hammerstein equation with smooth as well as weakly singular kernels. A fast algorithm is based upon the block truncation strategy which was recently proposed in [10]. A multilevel augmentation method for the linearized Hammerstein equation is subsequently proposed which further accelerates the solution process while maintaining the order of convergence. Numerical examples are given throughout this dissertation
    • …
    corecore