Multivariate Splines And Some Applications

Computational geometry is a new subject of geometry. It is a interdisciplinary subject of geometry, computational mathematics and computer science. The spline is a basic theoretical tool of computational geometry. After I. J. Schoenberg established the theory of the univariate spline in 1946, the research on the multivariate spline does not improve essentially. In 1975, Prof. Renhong Wang established the algebraic geometry method for the multivariate spline using the so-called conforamlity method of smoothing cofactor (CSC method). Then the sufficient and necessary conditions for multivariate splines over arbitrary partitions are obtained. The problems of multivariate splines over arbitrary partitions turn to solving a system of equations with polynomials as coefficients, and the theoretical framework of multivariate splines is established. This article uses the conformality method of smoothing cofactor to discuss the theories and applications of multivariate splines. The main work is as follows:1. We retrospect the basic theoretical framework of multivariate splines established by Prof. Renhong Wang, and discuss the conformality method of smoothing cofactor and his advanced products about the research on the space of multivariate splines.2. In 1980, under the basic theoretical framework of multivariate splines established by Prof. Renhong Wang, G. Farin considered the relationship between the barycentric coordinates of a piecewise polynomial over two adjacent triangles and smoothness con-ditions. He introduced the B-net method which is suitable for studying the multivariate splines over simplex partitions. We know that, the CSC method is an approach to study multivariate splines over arbitrary partition. So there is some relationship between the CSC method and the B-net method. This article indicates that the B-net method is established the basic theoretical framework of multivariate splines established by Prof. Renhong Wang. It could be derived directly by the CSC method.3. Scattered data approximation is a very important issue in computational geome-try. This article discusses the structure and nature of spline space (?). We proposed an algorithm (named BS2 Algorithm), which uses the bases in (?)to fitting the given scattered data points. The BS2 Algorithm avoids the selection of the Lagrange interpo-lation set. The numerical results indicate that the BS2 Algorithm has fine accuracy. In order to obtain a approximation function with higher accuracy and finer smoothness, we modify the BS2 Algorithm and propose a new algorithm:MBS Algorithm. It makes use of a hierarchy of 2-type partitions to generate a sequence of functions whose sum approaches the desired approximation functions.4. Using 2-type partitions spline space with boundary conditions, (?) and (?), this article proposes a numerical method to solve linear parabolic equations. We give an adaptive discontinuous Galerkin finite element method (the ADB method). The numerical results show that the solutions obtained by the ADB method have high accuracy. Since the bases of (?) and (?) are easy to store and evaluate, the ADB method is more convenient to implement.