有限元概率算法是应用蒙特卡罗方法去求偏微分方程有限元逼近的近似解。使用这种方法可以在不形成总刚矩阵的情况下直接地求出有限元解在某个点或少数几个点处的近似值，不但能节省计算机内存单元且程序易于实现。

近年来，有限元概率算法在国内外引起了不同的反响，有的高度评价这种新的算法，有的基本上否定这种算法。本书进一步讨论这一方法，得到了一些新的结果。

Foreward

前言

Chapter 1 Introduction ／ 1

Chapter 2 Brief Introduction of Markov Chain and Nonnegative Matrices ／ 6

Chapter 3 The Methods of Solving Elliptic Boundary Value Problem ／ 12

 3.1 Elliptic Boundary Value Prollem and Limit Transfer Matrix Q ／12

 3.2 Method of Solving Nonhomogeneous Elliptic Boundary Value Problem ／ 15

 3.3 Solving Parabolic Problem ／ 17

 3.4 Monte—Carlo Method of Computing Qand S ／19

 3.5 M ethods of Fast Approximate to limit M atrices Q and S ／ 22

 3.6 Under the Case That P Is Not Nonnegative Matrix／24

 3.7 Iterative Method for Finite Element Probability Computing／26

Chapter 4 The Finite Element Probability Computing Method / 28

Chapter 5 High Accuracy Methods of FiniteElement Probability Computing Method / 35

  5.1 The Probability Multigrid Method / 35

  5.2 The Boundary Thickening Method / 41

  5.3  Numerical Experiment / 42

Chapter 6 Rectangular Finite Element Probability Computing Method / 45

  6.1  Introduction / 45

  6.2 Probability Computing Model and Its Convergence Conditions / 46

  6.3  Numerical Experiment / 50

Chapter 7  Dimentional Independence / 52

Chapter 8 The Fast Computing Scheme of the Finite Element Method for the Two Point Boundary Problem / 57

  8.1  Model Problem and P'k-type Finite Element Space / 57

  8.2 The Probability Computing Scheme / 62

  8.3 Numerical Experiment / 65

Chapter 9 Example Analysis / 67

  9.1  The Monte-Carlo Method for Three-dimensional Problems / 67

  9.2 The Finite Element Monte-Carlo Method of Plate Problems / 69

Chapter 10 The Space Decomposition Method of the Finite Element / 75

  10.1  Abstract Problem / 75

  10.2  The Domain Decomposition Method and the Structure of Space S0 / 81

  10.3  Example / 83

  10.4 Probability Computing Method / 85

References / 87