Numerical Solution Of Partial Differential Equations By Bivariate Spline Finite Method

Multivariate splines are widely applied in approximation theory, computer aided geomet-ric design and finite element method. It is often used as the shape function space in solving differential equations by finite element method. The spline finite element method has many advantages such as the sparsity and symmetry of the coefficient matrix, the neglect of natural boundary conditions. But this method also has some disadvantages, for example, it can only be used in specular domains. For these reasons, The spline finite method can be considered as a supplement for application.In chapter 1, we introduce the smoothing cofactor-conformality method, B net method and the rudiments of finite element method, which would be used in the following work.In chapter 2, firstly the one dimensional cubic B spline is introduced. Secondly, we mention the space S21,0(Δmn(2)). Thirdly, Using cofactor-conformality method, a kind of cubic spline space with homogeneous boundary conditions S32,1,0(Δmn(2)) is constructed. At the end of this chapter, we present the exact expressions of two kinds of bivariate cubic splines with duplicated knots, which have the property of partition of unity.In chapter 3, we mainly solve partial differential equations by spline finite element method. On one hand, we solve second order elliptic and parabolic partial differential equations by the bi-variate cubic splines which are constructed in chapter 2, on the other hand, we solve biharmonic equations by quartic spline finite method. In theory, we could solve biharmonic equations by quadratic splines, but the tensor product cubic splines method is more widely used because of its fast convergency. We use quartic B spline finite method to get the numerical solution, examples show the efficiency of this method.