Bivariate Spline Method For Numerical Solution Of Partial Differential Equations

Spline is an excellent tool for numerical approximation, which is a perfect result of the combination of approximation theory and computer theory. In modern computational work, spline is the major tool for engineers and CAD software since it is easy to represent and efficient to evaluate. The spline finite element is also an efficient tool in finite element analysis. It is usually preferable in stress analysis of high building, because the splines use less degree of freedom than common finite elements.The main purpose of this thesis is to discuss about the spline method and its application in numerical solution of partial differential equations, which was originally proposed by Prof. Lai and his cooperators[42, 57, 58]. Actually, it can be viewed as a special spline finite element method. Brezzi and Fortin give an definition of the finite element method in [18] as "The finite element method is a general technique to build finite-dimensional subspaces of a Hilbert space V in order to apply the Ritz-Galerkin method to a variational problem." This is of course a very abstract way to define finite element method and this is for sure not the best way to understand it from the computational point of view, but it exactly conclude the basic idea of finite element method which includes large number of method. Let us take it as "abstract finite element method". At the same time, we refer the phrase "the standard finite element method(FEM)" as the concrete computational method which are as described as in the common textbooks[1, 45, 72, 80]. The spline method is regarded to be included in the framework of abstract finite element method. The main difference between them lies in the way of constructing the finite dimensional subspace of the Hilbert space. In FEM, this task is accomplished by an object named finite elements whose knows the basis of the finite element space. While in the spline method, the function spaces are represented in spline mode which do not required the construction of explicit finite elements.According to the relationship between the spline FEM and the standard FEM, the theory of standard FEM is also fit for spline method. Then it is sufficient to promise the existence of corresponding spline spaces. Thanks to the literatures of multivariate spline analysis, the existence theory of two dimension spline spaces is fruitful[63]. It is worth to point out that if the spline space of spline method and the finite element space of the FEM are same, then the approximated solutions are also the same. From this point of view, the spline method seems to have no more advantage than FEM. But from the success of the nonconforming finite element method, we can get some clue to get better approximation by spline method. The nonconforming finite elements benefit much form without imposing too much restriction on the element boundary. The spline method is also pay much attention to conformal condition on the element boundary, there seems to be some relationship between them. Although we would not concerned about nonconforming finite element method in this thesis, it really gives us many inspirations in treating the conforming/smoothness conditions of the spline spaces. We are intended to regard the spline method as a nonconforming way to treat conforming problem.The first chapter is concerned about the basic knowledge of the standard FEM and multivariate spline analysis. First, we introduce the motivation of current research on the spline method. For the reference of the remain chapters, some needed basic notations and results in FEM theory are presented in the second section. The basic knowledge about spline analysis is also introduced including some results on the existence theory and the approximation power of the spline spaces in the sense of Sobolev norms. The knowledge mentioned here is far away from the results supplied in the literatures, but they are picked out for quick reference.In the second chapter we introduce the triangle B-form and some of its useful algorithm on the triangle, such as the de casteljau algorithm, degree elevate algorithm, etc. Some of them can even accelerate discretization procedure remarkably. Next, the B-form representation of bivariate spline spaces are introduced as well as the conforming/smoothness condition of B-form on adjacent two triangles. Following the examples of dirichlet boundary value problems of second order and forth order, they are discretized into the following saddle point problem The spline method is well established through detail descriptions. The numerical examples give the most promising results to prove the effectiveness of the spline method.As we have seen in the second chapter, the spline method is very flexible which is the main reason for us being interested in. In the third chapter, we begin to explore this ability in p adaptive version of spline method. The adaptive FEM is an interested topic recently, however, there are still many unsettled problems in this area, one of which is the efficient local p-adaptivity. The FEM can not treat the local p-adaptivity of C~1 scheme yet. Since the spline method is powerful in construction of various spline spaces, it is a trivial task for treating the local p-adaptivity of the C~1 spline spaces. In this chapter, we first analyze the local conforming/smoothness condition for two adjacent triangle with different degrees, then it is applied to the standard spline method by only replacing the conforming condition with the new one. We also discuss about the detail of local adaptive techniques which are borrowed from the finite element method. At last the local h adaptive and p adaptive procedure are combined by hand, the spline method shows good ability in treating the elliptic problem with a singularity. To the best of our knowledge, the C~1 FEM can't treat the local p-adaptivity at present day.We continue on discussing the adaptivity of the spline method in the forth chapter. This time we start from the h-adaptivity. Noticing that the existence of hanging nodes can reduce the amount of elements remarkably, we emphasize on such case for the h-adaptive spline method. As in multivariate spline analysis, the most concerned problem is how to keep the conforming/smoothness condition across the element boundary with hanging nodes. We first archive the smoothness condition across the interface with arbitrary level hanging nodes, then the spline method is not difficult to implement according to this condition. A new method to solve the resulted constrained linear system is proposed in this chapter. It is proved to be an efficient way to solve such kind of linear system. We think it is important because this algorithm actually links the spline method with the standard FEM.In the last section, we try to apply the spline method to the bivariate Navier-Stokes problem. In the FEM literatures, it is preferred to solve this problem with mixed finite element method based on the velocity-pressure formulation. This may due to the difficulty in construction of C~1 finite element spaces. As we have seen, the spline method have no difficulties in implementation of C~1 version. So we adopt the stream function formulation of Navier-Stokes equation which reduce the degree of freedom remarkably. The newton iteration method and other numerical continuation techniques are incorporated into solving the ultimate nonlinear system. At last, we simulate three benchmark problems and get the desired result. They prove that the spline method is easy to manipulate and flexible in solving nonlinear partial differential equations.The current thesis investigate into the spline method in the following three aspects: Firstly, the smoothness condition across interface of two triangle pathes with different degrees, then the p-adaptivity is implemented easily with such conforming conditions, in which the C~1 adaptivity of the spline method is more competitive than FEM. Secondly, the h-adaptive spline method on the triangle mesh with arbitrary level hanging nodes is presented. By the way, the required continuity conforming condition across the interface with arbitrary level hanging nodes is proposed. The matrix modify method is applied to solve the resulting linear equations, and it is found to link between the spline method and FEM. At last, the spline method is applied to solve two dimensional incompressible problem. Thanks to the nice properties of B-form and the barycentric coordinate, the computational effort of the element matrix can be reduced remarkably corresponding to [57]. The spline method was proposed very recently and there are many application which have not been study, we would keep on investigating this area.