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Elliptic equations in two dimensions:


Suppose that R is a bounded region in the $ (x_{1},x_{2})$ plane with boundary $ \partial R$. The equation

(30)

is said to be elliptic in R if $ b^{2}-ac < 0$ for all points $ (x_{1},x_{2})$ in R. Three distinct problems involving equation (30) arise depending on the boundary conditions prescribed on $ \partial R:$
  1. The first boundary value problem or Dirichlet problem, requires a solution u of equation (30) which takes on prescribed values

    $ (x_{1},x_{2})$ (31)

    on the boundary $ \partial R.$
  2. The second boundary value problem or Neumann problem, where

    $\displaystyle \frac{\partial u}{\partial x}=g(x_{1},x_{2}))$ (32)

    on the boundary $ \partial R$. Here $ \displaystyle \frac{\partial}{\partial x}$ refers to derivative along the normal to $ \partial R$ directed away from the interior of $ R$.
  3. The third boundary value problem or Robbins problem, with

    $\displaystyle \alpha(x_{1},x_{2})u+\beta(x_{1},x_{2})\frac{\partial u}{\partial x}=\gamma(x_{1},x_{2})$ (33)

    on $ \partial R,$ where $ \alpha (x_{1},x_{2})>0,\beta(x_{1},x_{2})>0$ for $ (x_{1},x_{2})\epsilon \partial R.$
Before developing finite difference methods of solving elliptic equations, a most useful analytical tool in the study of elliptic partial differential equation will be introduced. This is the Maximum Principle which will be stated for the linear elliptic equation.

where a,b,c,d and e are functions of the independent variables $ x_{1},x_{2}$. It is clear that in this case any constant represents a solution of the equation. The maximum principle states that the constants are the only solutions which can assume a maximum or minimum value in the interior of the bounded region R. Alternatively, it states that every solution of the elliptic equation achieves its maximum and minimum values on the boundary $ \partial R$ of R.

Laplace's equation in a square:
We consider Laplace's equation

$\displaystyle \frac{\partial^{2}u}{\partial x^{2}_{1}}+\frac{\partial^{2}u}{\partial x^{2}_{2}}=0$ (34)

subject to $ u=f(x_{1},x_{2})$ on the boundary of the unit square $ 0\leq x_{1},x_{2}\leq1$. The square region is covered by a grid with sides parallel to the coordinate axes and the grid spacing is $ h$. If $ Mh=1$, the number of internal grid points or nodes is $ (M-1)^{2}$. The coordinates of a typical internal grid point are , $ X_{2}=mh$ ( and m integers ) and the value of $ u$ at this grid point is denoted by $ u_{l,m}$. Using Taylor's Theorem, we obtain

and

$\displaystyle u_{l-1,m}=(u-h\frac{\partial u}{\partial x_{1}}+ \frac{1}{2}h^{2}... ...}^{3}}+ \frac{1}{24}h^{4}\frac{\partial^{4}u}{\partial x_{1}^{4}}+......)_{l,m}$

and after addition

$\displaystyle u_{l+1,m}+u_{l-1,m}-2u_{l,m}=(h^{2}\frac{\partial^{2}u}{\partial ... ..._{1}}+ \frac{1}{12}h^{4}\frac{\partial^{4}u}{\partial x_{1}^{4}}+.......)_{l,m}$

Similarly.

$\displaystyle u_{l,m+1}+u_{l,m-1}-2u_{l,m}=(h^{2}\frac{\partial^{2}u}{\partial ... ..._{2}}+ \frac{1}{12}h^{4}\frac{\partial^{4}u}{\partial x_{2}^{4}}+.......)_{l,m}$

and so

$\displaystyle u_{l+1,m}+u_{l-1,m}+u_{l,m+1}+u_{l,m-1}-4u_{l,m}= \left[ h^{2}\le... ..._{1}^{4}}+\frac{\partial^{ 4}u}{\partial x_{2}^{ 4}}\right)+......\right]_{l,m}$

leading to the five point finite difference scheme

$\displaystyle U_{l+1,m}+U_{l-1,m}+U_{l,m+1}+U_{l,m-1}-4U_{l,m}=0$ (35)

for the Laplace's equation, with a local truncation error

$\displaystyle \frac{1}{12}h^{4}\left(\frac{\partial^{4}u}{\partial x^{4}_{1}}+ \frac{\partial^{4}u}{\partial x^{4}_{2}}\right)_{l,m}+.......$

where $ U_{l,m}$ denotes the function satisfying the difference equation at the grid point

$\displaystyle X_{1}=lh,X_{2}=mh$

The totality of equations at the $ (M-1)^{2}$ internal grid points of the unit square leads to the matrix equation

$\displaystyle AU=K$ (36)

where $ A$ is a matrix of order $ (M-1)^{2}$ given by


with $ J$ the unit matrix of order $ (M-1)$ and $ B$ a matrix of order

$ (M-1)$ given by

B =

The vectors $ U$ and $ K$ are given by

U= $\displaystyle \{U_{1,1},....,U_{1,M-1};U_{2,1},....,U_{2,M-1};........;U_{M-1,1},.....,U_{M-1,M-1}\}^{T}$

and

respectively, where $ \{\}^{T}$ denotes the transpose. The elements of the vector U constitute the $ (M-1)^{2}$ unknowns $ U_{l,m}(1\leq l,m,M-1)$ and the elements of the vector K depend on the boundary values $ f(x_{1},x_{2})$ at the grid points on the perimeter of the unit square. Because of the large number of zero element in the matrix A, iterative methods are often used to solve the system (36).

 

Exercises

 

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