Probabilistic Numerical Method For The Dirichlet

The relationship and interplay between probability theory and analysis is remarkable, for example, the connections between diffusion processes and second-order elliptic partial differential equations. Based on these, a new numerical method for the Dirichlet problem is proposed in this thesis. This method overcomes the difficulties and limitations resulting from others. It shows that solving problems in analysis by probability theory has a bright future both in theory and practice.It is usually that the solution for the Dirichlet problem can not be obtained actually. With the development of computers, people pay more and more attention to the numerical solution. Many ideas and methods have appeared. Among them the finite element method has become most popular. But the finite element method has some weaknesses and limitations too. First, it is not fit to higher-dimensional problems. For instance, about 3-dimensional problems there exist many difficulties in the shape of the element, the subdivison and the solution of the equations. Especially, the unknowns of the algebraic equations about 3-dimensional problems increase as geometrical progression compared with 2-dimensiond problems. So development of computers can not meet their needs. Second, the computational amount is huge and unnecessary when it is used to solve the problem of concentrative loading in mechanics and physics. Third, it can not solve the exterior Dirichlet problem. Therefore, based on probability theory and its links with the Dirichlet problem, we propose a new numerical method for the Dirichlet problem to remove the above weaknesses.Its main idea is as follows: Firstly, represent the solution of the Dirichlet problem as the stochastic representations. Then subdivide the boundary of the domain and make the problem discretized. Sequently, use the strong Markov property and the distributions of the time and place of hitting spheres for Brownian motion or Brownian motion with drift, and construct an auxiliaryball for the domain. Finally get the solution to the discretized problem. So the problem on the domain is turned into a problem on the boundary and the 3-D problem is easily solved when it becomes a 2-D problem. Because the numerical solution is obtained point to point by it, the probabilistic numerial method is not expensive for the problem of concentrative loading. For the stochasic representation is still correct over unbounded domains, this method can also solve the exterior Dirichlet problem. Last, the method is adaptable to boundary and can be used to various problems with complex boundary.This thesis consists of seven chapters.In Chapter 1, the present conditions of numerical methods for the Dirichlet problem, the probability theory which connects with the Dirichlet problem, and the essence and characteristics of the probabilistic numerical method are summarized. In chapter 2, the essential conceptions, properties and theorems etc. in the probability theory which is involved in the thesis are introduced. The remainder of the paper falls into two parts. The first part, which composed of chapter 3, 4, 5 and chapter 6, is devoted to studying the new probabilistic numerical method. Chapter 3 introduces the main ideas of the probabilistic numerical method and applies it to the harmonic boundary value problem over bounded plane domains. The numerical examples show the method is both covenient and efficient. Chapter 4 generalizes the method to higher-dimensional problems, the Dirichlet-Poisson problem and general problems. The method is verified effective again through numerical examples. Chapter 5 establishes the probabilistic numerical method for the Dirichlet problem relevant to Brownian motion with drift. In chapter 6 the method is applied to the exterior Dirichlet problem and its another advantage is given. The second part contains chapter 7, which is contributed to studying other main numerical methods based on probability theory: Monte-Carlo finite difference method and Monte-Carlo finite element method.