| Boundary value problems for elliptic partial differential equations of second order are appeared in many fields such as physics,mechanics and etc. It is usually that the solution for the problem can not be easily obtained actually because of complex boundary or unbounded domains. With the development of computers, people pay more and more attention to the numerical solution. Both in theory and practice,it has a bright future to develope new methods and improve traditional methods.A efficient probabilistic computing method for harmonic and Poisson problems is proposed. In the thesis this method is further studied and is extended to a kind of complicated elliptic boundary value problems over bounded and unbounded domains. Based on the remarkable relationship between diffusion processes and second-order elliptic partial differential equations, we firstly represent the solution of the boundary value problem as the expectation of diffusion processes to make the problem discretized. Finally, by using the strong Markov property of Brownian motion etc to obtain the solutions to the discretized problems. The study show the method has some advantages compared with some others.In Chapter 1, the present conditions of numerical methods for the elliptic problems, the probability theory which connects with the elliptic equations, and the main numerical methods are summarized. Especially the main ideas of the probabilistic numerical method are introduced.In chapter 2, the essential conceptions, properties and theorems etc. in the elliptic partial differential equations and probability theory involved in the thesis are introduced.Chapter 3 establishes the probabilistic numerical method for a kind of boundary value problems over bounded domains.Chapter 4 generalizes the method to a kind of complicated boundary value problems over unbounded domains. It bases on the stochastic representations of solutions. In the bounded domain outside the unbounded,an auxiliary ball is constructed to make the problem over unbounded domaims turned into a problem over boundary. Subdivision over boundary is needed to make the problem discretized. The distributions of the time and place of hitting spheres for Brownian motion or Brownian motion with drift from outside spheres is emplored to gain the numerical solution.In Chapter 5, the numerical example shows the probabilistic method is both covenient and efficient. Finally,we summary briefly the method. |
