Finite Difference Methods for the Poisson Equation#
This notebook will focus on numerically approximating a inhomogenous second order Poisson Equation.
The Differential Equation#
The general two dimensional Poisson Equation is of the form:
with boundary conditions
Homogenous Poisson Equation#
This notebook will implement a finite difference scheme to approximate the inhomogenous form of the Poisson Equation \(f(x,y)=100(x^2+y^2)\):
with the Boundary Conditions:
# LIBRARY
# vector manipulation
import numpy as np
# math functions
import math
# THIS IS FOR PLOTTING
%matplotlib inline
import matplotlib.pyplot as plt # side-stepping mpl backend
import warnings
warnings.filterwarnings("ignore")
from IPython.display import HTML
from mpl_toolkits.mplot3d import axes3d
import matplotlib.pyplot as plt
Discete Grid#
The region \(\Omega=(0,1)\times(0,1)\) is discretised into a uniform mesh \(\Omega_h\). In the \(x\) and \(y\) directions into \(N\) steps giving a stepsize of
resulting in
and
The Figure below shows the discrete grid points for \(N=10\), the known boundary conditions (green), and the unknown values (red) of the Poisson Equation.
N=10
h=1/N
x=np.arange(0,1.0001,h)
y=np.arange(0,1.0001,h)
X, Y = np.meshgrid(x, y)
fig = plt.figure()
plt.plot(x[1],y[1],'ro',label='unknown');
plt.plot(X,Y,'ro');
plt.plot(np.ones(N+1),y,'go',label='Boundary Condition');
plt.plot(np.zeros(N+1),y,'go');
plt.plot(x,np.zeros(N+1),'go');
plt.plot(x, np.ones(N+1),'go');
plt.xlim((-0.1,1.1))
plt.ylim((-0.1,1.1))
plt.xlabel('x')
plt.ylabel('y')
plt.gca().set_aspect('equal', adjustable='box')
plt.legend(loc='center left', bbox_to_anchor=(1, 0.5))
plt.title(r'Discrete Grid $\Omega_h,$ h= %s'%(h),fontsize=24,y=1.08)
plt.show();
Boundary Conditions#
The discrete boundary conditions are
The Figure below plots the boundary values of \(w[i,j]\).
w=np.zeros((N+1,N+1))
for i in range (0,N):
w[i,0]=np.sin(2*np.pi*x[i]) #left Boundary
w[i,N]=np.sin(2*np.pi*x[i]) #Right Boundary
for j in range (0,N):
w[0,j]=2*np.sin(2*np.pi*y[j]) #Lower Boundary
w[N,j]=2*np.sin(2*np.pi*y[j]) #Upper Boundary
fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
# Plot a basic wireframe.
ax.plot_wireframe(X, Y, w,color='r', rstride=10, cstride=10)
ax.set_xlabel('x')
ax.set_ylabel('y')
ax.set_zlabel('w')
plt.title(r'Boundary Values',fontsize=24,y=1.08)
plt.show()
Numerical Method#
The Poisson Equation is discretised using \(\delta_x^2\) is the central difference approximation of the second derivative in the \(x\) direction
and \(\delta_y^2\) is the central difference approximation of the second derivative in the \(y\) direction
The gives the Poisson Difference Equation,
where \(w_ij\) is the numerical approximation of \(U\) at \(x_i\) and \(y_j\). Expanding the the Poisson Difference Equation gives the five point method,
for \(i=1,...,N-1\) and \(j=1,...,N-1.\)
Matrix form#
This can be written as a system of \((N-1)\times(N-1)\) equations can be arranged in matrix form
where \(A\) is an \((N-1)^2\times(N-1)^2\) matrix made up of the following block tridiagonal structure
where \(I\) denotes an \(N-1 \times N-1\) identity matrix and \(T\) is the tridiagonal matrix of the form:
The plot below shows the matrix \(A\) and its inverse \(A^{-1}\) as a colourplot.
N2=(N-1)*(N-1)
A=np.zeros((N2,N2))
## Diagonal
for i in range (0,N-1):
for j in range (0,N-1):
A[i+(N-1)*j,i+(N-1)*j]=-4
# LOWER DIAGONAL
for i in range (1,N-1):
for j in range (0,N-1):
A[i+(N-1)*j,i+(N-1)*j-1]=1
# UPPPER DIAGONAL
for i in range (0,N-2):
for j in range (0,N-1):
A[i+(N-1)*j,i+(N-1)*j+1]=1
# LOWER IDENTITY MATRIX
for i in range (0,N-1):
for j in range (1,N-1):
A[i+(N-1)*j,i+(N-1)*(j-1)]=1
# UPPER IDENTITY MATRIX
for i in range (0,N-1):
for j in range (0,N-2):
A[i+(N-1)*j,i+(N-1)*(j+1)]=1
Ainv=np.linalg.inv(A)
fig = plt.figure(figsize=(12,4));
plt.subplot(121)
plt.imshow(A,interpolation='none');
clb=plt.colorbar();
clb.set_label('Matrix elements values');
plt.title('Matrix A ',fontsize=24)
plt.subplot(122)
plt.imshow(Ainv,interpolation='none');
clb=plt.colorbar();
clb.set_label('Matrix elements values');
plt.title(r'Matrix $A^{-1}$ ',fontsize=24)
fig.tight_layout()
plt.show();
The vector \(\mathbf{w}\) is of length \((N-1)\times(N-1)\) which made up of \(N-1\) subvectors \(\mathbf{w}_j\) of length \(N-1\) of the form
The vector \(\mathbf{r}\) is of length \((N-1)\times(N-1)\) which made up of \(N-1\) subvectors of the form \(\mathbf{r}_j=-h^2\mathbf{f}_j-\mathbf{bx}_{j}-\mathbf{by}_j\), where \(\mathbf{bx}_j \) is the vector of left and right boundary conditions
for \(j=1,..,N-1\), where \(\mathbf{by}_j\) is the vector of the lower boundary condition for \(j=1\),
upper boundary condition for \(j=N-1\)
for \(j=2,...,N-2\) \begin{equation}\mathbf{by}_j=0,\end{equation} and
for \(j=1,...,N-1\).
r=np.zeros(N2)
# vector r
for i in range (0,N-1):
for j in range (0,N-1):
r[i+(N-1)*j]=100*h*h*(x[i+1]*x[i+1]+y[j+1]*y[j+1])
# Boundary
b_bottom_top=np.zeros(N2)
for i in range (0,N-1):
b_bottom_top[i]=np.sin(2*np.pi*x[i+1]) #Bottom Boundary
b_bottom_top[i+(N-1)*(N-2)]=np.sin(2*np.pi*x[i+1])# Top Boundary
b_left_right=np.zeros(N2)
for j in range (0,N-1):
b_left_right[(N-1)*j]=2*np.sin(2*np.pi*y[j+1]) # Left Boundary
b_left_right[N-2+(N-1)*j]=2*np.sin(2*np.pi*y[j+1])# Right Boundary
b=b_left_right+b_bottom_top
Results#
To solve the system for \(\mathbf{w}\) invert the matrix \(A\)
such that
Lastly, as \(\mathbf{w}\) is in vector it has to be reshaped into grid form to plot.
The figure below shows the numerical approximation of the homogeneous Equation.
C=np.dot(Ainv,r-b)
w[1:N,1:N]=C.reshape((N-1,N-1))
fig = plt.figure(figsize=(8,6))
ax = fig.add_subplot(111, projection='3d');
# Plot a basic wireframe.
ax.plot_wireframe(X, Y, w,color='r');
ax.set_xlabel('x');
ax.set_ylabel('y');
ax.set_zlabel('w');
plt.title(r'Numerical Approximation of the Poisson Equation',fontsize=24,y=1.08);
plt.show();
Consistency and Convergence#
We now ask how well the grid function determined by the five point scheme approximates the exact solution of the Poisson problem.
Consistency#
Consitency (Definition)#
Let
denote the finite difference approximation associated with the grid \(\Omega_h\) having the mesh size \(h\), to a partial differential operator
a simply connected, open set \(\Omega \subset R^2\). For a given function \(\varphi\in C^{\infty}(\Omega)\), the truncation error of \(\nabla^2_h\) is
The approximation \(\nabla^2_h\) is consistent with \(\nabla^2\) if
for all \(\mathbf{x} \in D\) and all \(\varphi \in C^{\infty}(\Omega)\). The approximation is consistent to order \(p\) if \(\tau_h(\mathbf{x})=O(h^p)\).
In other words a method is consistent with the differential equation it is approximating.
Proof of Consistency#
The five-point difference analog \(\nabla^2_h\) is consistent to order 2 with \(\nabla^2\).
Proof
Pick \(\varphi \in C^{\infty}(D)\), and let \((x,y) \in \Omega\) be a point such that \((x\pm h, y),(x,y \pm h) \in \Omega\bigcup \partial\Omega\). By the Taylor Theorem
where \(\zeta^{\pm} \in (x-h,x+h)\). Adding this pair of equation together and rearranging , we get
By the intermediate value theorem
for some \(\zeta \in (x-h,x+h)\). Therefore,
Similar reasoning shows that
for some \(\eta \in (y-h,y+h)\). We conclude that \(\tau_h(x,y)=(\nabla-\nabla_h)\varphi(x,y)=O(h^2).\)
Convergence#
Definition#
Let \(\nabla^2_hw(\mathbf{x}_j)=f(\mathbf{x}_j)\) be a finite difference approximation, defined on a grid mesh size \(h\), to a PDE \(\nabla^2U(\mathbf{x})=f(\mathbf{x})\) on a simply connected set \(D \subset R^n\). Assume that \(w(x,y)=U(x,y)\) at all points \((x,y)\) on the boundary \(\partial\Omega\). The finite difference scheme converges (or is convergent) if
Theorem (DISCRETE MAXIMUM PRINCIPLE).#
If \(\nabla^2_hV_{ij}\geq 0\) for all points \((x_i,y_j) \in \Omega_h\), then
If \(\nabla^2_hV_{ij}\leq 0\) for all points \((x_i,y_j) \in \Omega_h\), then
Propositions#
The zero grid function for which \(U_{ij}=0\) for all \((x_i,y_j) \in \Omega_h \bigcup \partial\Omega_h\) is the only solution to the finite difference problem
For prescribed grid functions \(f_{ij}\) and \(g_{ij}\), there exists a unique solution to the problem
Definition#
For any grid function \(V:\Omega_h\bigcup\partial\Omega_h \rightarrow R\),
Lemma#
If the grid function \(V:\Omega_h\bigcup\partial\Omega_h\rightarrow R\) satisfies the boundary condition \(V_{ij}=0\) for \((x_i,y_j)\in \partial\Omega_h\), then
Given these Lemmas and Propositions, we can now prove that the solution to the five point scheme \(\nabla^2_h\) is convergent to the exact solution of the Poisson Equation \(\nabla^2\).
Convergence Theorem#
Let \(U\) be a solution to the Poisson equation and let \(w\) be the grid function that satisfies the discrete analog
Then there exists a positive constant \(K\) such that
where
Proof
The statement of the theorem assumes that \(U\in C^4(\bar{\Omega})\). This assumption holds if \(f\) and \(g\) are smooth enough. \begin{proof} Following from the proof of the Proposition we have
for some \(\zeta \in (x_{i-1},x_{i+1})\) and \(\eta_j\in(y_{j-1},y_{j+1})\). Therefore,
If we subtract from this the identity equation \(-\nabla_h^2w_{ij}=f_{ij}\) and note that \(U-w\) vanishes on \(\partial\Omega_h\), we find that
It follows that