This paper continues the theme of the recent work [3] and further develops the Petrov-Galerkin method for Fredholm integral equations of the second kind. Specifically, we study wavelet Petrov-Galerkin schemes based on discontinuous orthogonal multiwavelets and prove that the condition number of the coefficient matrix for the linear system obtained from the wavelet Petrov–Galerkin scheme is bounded. In addition, we propose a truncation strategy which forms a basis for fast wavelet algorithms and analyze the order of convergence and computational complexity of these algorithms.

The study of operator equations is an important branch of mathematics. The fundamental theory of operator equations, linear or nonlinear, formulated in a Hilbert or a Banach space setting, is originated from the classical theory of differential and integral equations. This modern theory of differential and integral equations has been well developed in the last few decades, in which many profound concepts, results, methods, and algorithms were established with considerable generality. On the one hand, various types of mathematical equations, such as linear and nonlinear differential, integral, integro-differential, and functional equations, can all be unified under the same framework of abstract operator equations. On the other hand, many well-known theories and methods in Functional Analysis and Operator Theory have proven very effective and useful in the study of basic solvability problems in operator equations, including not only the existence and uniqueness of a solution but also efficient numerical algorithms for approximating the solution. The theory and methodology of operator equations have now played a very important role in computational mathematics, applied sciences and engineering.

We develop in this paper a theoretical framework for the analysis of convergence for the Petrov-Galerkin method and superconvergence for the iterated Petrov-Galerkin method for Fredholm integral equations of the second kind. As important approaches to the analysis, we introduce notions of the generalized best approximation and the regular pair of trial space sequence and test space sequence. In Hilbert spaces, we characterize the regular pair in terms of the angle of two space sequences or the generalized best approximation projections. Several specic constructions of the Petrov-Galerkin elements for equations of both one dimension and two dimensions are presented and the convergence of the Petrov-Galerkin method and the iterated Petrov-Galerkin method using these elements is proved.

We introduce the concept of a re nable set relative to a family of contractive mappings on a metric space, and demonstrate how such sets are useful to recursively construct interpolants which have a multiscale structure. The notion of a re nable set parallels that of a re nable function, which is the basis of wavelet construction. The interpolation points we recursively generate from a re nable set by a set-theoretic multiresolution are analogous to multiresolution for functions used in wavelet construction. We then use this recursive structure for the points to construct multiscale interpolants. Several concrete examples of re nable sets which can be used for generating interpolatory wavelets are included.

In this paper, we develop a discrete wavelet Petrov–Galerkin method for integral equations of the second kind with weakly singular kernels suitable for solving boundary integral equations. A compression strategy for the design of a fast algorithm is suggested. Estimates for the rate of convergence and computational complexity of the method are provided.

In this note, the optimal L2-error estimate of the finite volume element method (FVE) for elliptic boundary value problem is discussed. It is shown that ‖u－uh‖0≤Ch2|lnh|1/2‖f‖1,1 and ‖u　uh‖0≤Ch2‖f‖1,p, p>1, where u is the solution of the variational problem of the second order elliptic partial differential　equation, uh is the solution of the FVE scheme for solving the problem, and f is the given　function in the right-hand side of the equation.

In this paper we develop fast collocation methods for integral equations of the second kind with weakly singular kernels. For this purpose, we construct multiscale interpolating functions and collocation functionals having vanishing moments. Moreover, we propose a truncation strategy for the coefficient matrix of the corresponding discrete system which forms a basis for fast algorithms. An optimal order of convergence of the approximate solutions obtained from the fast algorithms is proved and the computational complexity of the algorithms is estimated. The stability of the numerical method and the condition number of the truncated coefficient matrix are analyzed.

We introduce multilevel augmentation methods for solving operator equa-tions based on direct sum decompositions of the range space of the operator and the solution space of the operator equation and a matrix sphtting scheme. We establish a general setting/or the analysis of these methods, showing that the methods yield ap-proximate solutions ol the same convergence order as the best approximation from the subspace. These augmentation methods allow US to develop, fast, accurate and stable nonconventional numerical algorithms for solving operator equations. In particular. For second kind equations, special splitting techniques are proposed to develop such algo-rithms. These algorithms are then applied to solve the linear systems resulting form matrix compression schemes using wavelet-like functions|or solving Fredholm integral equations of the second kind. For this special case. a complete analysis for computa-tional complexity and convergence order is presented. Numerical examples are included to demonstrate the efficiency and accuracy of the methods. In these examples we use the proposed augmentation method to solve large scale linear systems resulting form the recently developed wavelet Galerkin methods and fast collocation methods applied to integral equations of the second kind. Our numerical results confirm that this aug-mentation method is particularly efficient for solving large scale linear systems induced from wavelet compression schemes.

We introduce a multilevel augmentation method for solving ill-posed operator equations by making use of the multiscale structure of the matrix representation of the operator. The method leads to fast solutions of the discrete regularization methods for the equations. Choices for a priori and a posteriori regularization parameters are proposed. An optimal convergence order for the method with the choices of parameters is established. Numerical results are presented to illustrate the accuracy and efficiency of the method.