# 利用最大值原理节省计算时间和资源

2017年 5月 9日

### 最大值原理简介

COMSOL Multiphysics 提供了仅在边界上查找极值的方法，并允许我们将解存储在有限的一组域或边界上。

### 推导最大值原理

\frac{d^2u}{dx^2}=0

u(x) = ax+b,

#### 具有均匀、各向同性性质的对流扩散问题

-\kappa\Delta u + \mathbf{v}\cdot \nabla u = f, \quad f<0, \textrm{ 在 } \Omega \textrm{中}

-\kappa\Delta u + \mathbf{v}\cdot \nabla u \ge 0

f<0

#### 各向异性，非均质相对流扩散问题

\nabla \cdot (-\mathcal{D} \nabla u + \mathbf{v}u) = f,

1. 扩散张量 D 是正定的
2. 对流速度 v 是无发散的；即 \nabla \cdot \mathbf{v} = 0
3. 源项为非正数；即 f(\mathbf{x}) \leq 0, \quad \forall \mathbf{x} \in \Omega

\nabla \cdot (-\mathcal{\kappa} \nabla T ) = \sigma|\nabla V|^2,

#### 瞬态问题

\frac{\partial u}{\partial t}+\nabla \cdot (-\mathcal{D} \nabla u +\mathbf{v}u) = f

\frac{\partial u}{\partial t}+\frac{\partial}{\partial x}(-D \frac{\partial u}{\partial x} +vu) = f, \quad x \in [0,L], \quad t\in[0,T], u(x,0) = u_0(x)

\frac{\partial u}{\partial t}-\frac{\partial D}{\partial x}\frac{\partial u}{\partial x} -D \frac{\partial^2 u}{\partial x^2} +v\frac{\partial u}{\partial x}+u\frac{\partial v}{\partial x} = f, \quad x \in [0,L], \quad t\in[0,T], u(x,0) = u_0(x)

– D \frac{\partial^2 u}{\partial x^2} = f

\frac{\partial u}{\partial t}- D \frac{\partial^2 u}{\partial x^2} = f \Rightarrow \frac{\partial u}{\partial t}= f+D \frac{\partial^2 u}{\partial x^2}

#### 系统和高阶方程

\sigma_{ij,j}+f_i = 0, i,j=1,2,3,

\sigma_{ij}=\lambda \epsilon_{kk}\delta_{ij}+2\mu \epsilon_{ij}

\qquad \epsilon_{ij}=\frac{1}{2}(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i})

\lambda \epsilon_{kk,i}+2\mu\epsilon_{ij,j}=0, \quad i,j=1,2,3

\lambda \epsilon_{kk,ii}+2\mu\epsilon_{ij,ji}=0, \quad i,j=1,2,3.

\epsilon_{kk,ii}=\Delta \epsilon_{kk} = 0

\frac{\partial^2 \phi}{\partial x^2}+\frac{\partial^2 \phi}{\partial y^2} =-2, \quad (x,y) \in \Omega

\sigma_{xz}=\mu\theta\frac{\partial \phi}{\partial y}, \quad \sigma_{yz}=-\mu\theta\frac{\partial \phi}{\partial x}

\tau = \sqrt{\sigma_{xz}^2+\sigma_{yz}^2} \Rightarrow \tau^2= \sigma_{xz}^2+\sigma_{yz}^2=(\mu\theta)^2|\nabla\phi|^2

div(|\nabla\phi|^2)=\frac{\partial^2}{\partial x_j\partial x_j}(\frac{\partial \phi}{\partial x_i}\frac{\partial \phi}{\partial x_i})=2\frac{\partial}{\partial x_j}(\frac{\partial \phi}{\partial x_i}\frac{\partial^2\phi}{\partial x_i\partial x_j})=2\frac{\partial^2\phi}{\partial x_i\partial x_j}\frac{\partial^2\phi}{\partial x_i\partial x_j}+2\frac{\partial\phi}{\partial x_i}\frac{\partial^3\phi}{\partial x_j\partial x_i\partial x_j}

div(|\nabla\phi|^2)=2\frac{\partial^2\phi}{\partial x_i\partial x_j}\frac{\partial^2\phi}{\partial x_i\partial x_j}+2\frac{\partial\phi}{\partial x_i}\frac{\partial}{\partial x_i}(\frac{\partial^2\phi}{\partial x_j\partial x_j})

div(\tau^2)>0 \quad \forall (x,y) \in \Omega

### 如何利用 COMSOL Multiphysics® 软件中的最大值原理

#### 边界元法

COMSOL Multiphysics 主要基于有限元方法。对于某些问题，它支持边界元方法。在最新版本的软件(5.3 版)中，边界元方法可用于静电、腐蚀和用户定义的方程。

### 关于使用最大值原理的结论性思考

• Lawrence C. Evans, Partial Differential Equations, American Mathematical Society, 2010
• Rene Sperb, Maximum Principles and Their Applications, Academic Press, Inc., 1981