 # L2 error increased with increase mesh size in poissoon equation

Hi all. I want to know that why L2 error increases with increase in mesh size if I am using Lagrange P2 element . Whether I am doing something wrong. The code which is given below I am using this one and get 2.952056 error with mesh size 1/2 and 7.1425555 with mesh size 1/16. Can someone please tell me why I am getting this much error.

``````from __future__ import print_function
from fenics import *
import matplotlib.pyplot as plt

# Create mesh and define function space
mesh = UnitSquareMesh(2,2)
V = FunctionSpace(mesh, 'P', 2)

# Define boundary condition
u_D = Expression('1 + x*x + 2*x*x', degree=2)

def boundary(x, on_boundary):
return on_boundary

bc = DirichletBC(V, u_D, boundary)

# Define variational problem
u = TrialFunction(V)
v = TestFunction(V)
f = Constant(-6.0)
L = f*v*dx

# Compute solution
u = Function(V)
solve(a == L, u, bc)

# Plot solution and mesh
plot(u)
plot(mesh)

# Save solution to file in VTK format
vtkfile = File('poisson/solution.pvd')
vtkfile << u

# Compute error in L2 norm
error_L2 = errornorm(u_D, u, 'L2')

# Compute maximum error at vertices
vertex_values_u_D = u_D.compute_vertex_values(mesh)
vertex_values_u = u.compute_vertex_values(mesh)
import numpy as np
error_max = np.max(np.abs(vertex_values_u_D - vertex_values_u))

# Print errors
print('error_L2  =', error_L2)
print('error_max =', error_max)

# Hold plot
plt.show()
``````

I can not reproduce this behaviour. When executing your code, I get values for error_L2 somewhere around 1e-15, which is practically zero, no matter the mesh size. This is expected with your chosen exact solution, as it is contained in your finite element function space.

Actually I want to solve the following problem:

and I did the following and run and got error_L2 = 0.14162139723192527
with mesh size 1/2 and error_L2 = 0.48687396343176065 with mesh size 1/8. Thats what I was saying L2 error increases with finer and finer mesh. Kindly help me regarding the same. Thanku.

``````
from __future__ import print_function
from fenics import *
import matplotlib.pyplot as plt

# Create mesh and define function space
mesh = UnitSquareMesh(4,4)
V = FunctionSpace(mesh, 'Lagrange', 2)

# Define boundary condition
u_D = Expression('sin(pi*x)*sin(pi*x)*sin(pi*x)*sin(pi*x) + cos(pi*x)*cos(pi*x)', degree=2)

def boundary(x, on_boundary):
return on_boundary

bc = DirichletBC(V, u_D, boundary)

# Define variational problem
u = TrialFunction(V)
v = TestFunction(V)
f = Expression('-8*pi*pi*pi*pi*sin(pi*x)*sin(pi*x)*cos(2*pi*x) - 8*pi*pi*pi*pi*sin(pi*x)*sin(pi*x)*cos(2*pi*x) + 8*pi*pi*pi*pi*cos(2*pi*x)*cos(2*pi*x) \
+ 4 *pi*pi*pi*pi*cos(pi*x)*cos(pi*x)', degree=0)
n  = FacetNormal(mesh)
h_avg = (h('+') + h('-'))/2

# Parameters
alpha = 6

# Bilinear form
#print('a=',a)

L = f*v*dx

# Compute solution
u = Function(V)
solve(a == L, u, bc)

# Plot solution and mesh
plot(u)
plot(mesh)

# Save solution to file in VTK format
vtkfile = File('biharmonic/solution.pvd')
vtkfile << u

# Compute error in L2 norm
error_L2 = errornorm(u_D, u, 'L2')

# Compute maximum error at vertices
vertex_values_u_D = u_D.compute_vertex_values(mesh)
vertex_values_u = u.compute_vertex_values(mesh)
import numpy as np
error_max = np.max(np.abs(vertex_values_u_D - vertex_values_u))

# Print errors
print('error_L2  =', error_L2)
print('error_max =', error_max)

# Hold plot
plt.show()
``````

I again did for poisson problem but got the same . Kindly tell me why this is happend . Error must be reduced by doing finner mesh but I got opposite of it. Kindly help me. Thanku.

``````ravina@ravina:~/Downloads\$ python3 poisson2.py
Solving linear variational problem.
*** Warning: Degree of exact solution may be inadequate for accurate result in errornorm.
error_L2  = 2.9520565726363387e-15(with mesh size(1/2))
error_max = 6.661338147750939e-16
Solving linear variational problem.
*** Warning: Degree of exact solution may be inadequate for accurate result in errornorm.
error_L2  = 5.603873381595536e-15(with mesh size 1/4)
error_max = 5.329070518200751e-15
``````
``````from __future__ import print_function
from fenics import *
import matplotlib.pyplot as plt

# Create mesh and define function space
mesh = UnitSquareMesh(4,4)
V = FunctionSpace(mesh, 'P', 2)

# Define boundary condition
u_D = Expression('1 + x*x + 2*x*x', degree=2)

def boundary(x, on_boundary):
return on_boundary

bc = DirichletBC(V, u_D, boundary)

# Define variational problem
u = TrialFunction(V)
v = TestFunction(V)
f = Constant(-6.0)
L = f*v*dx

# Compute solution
u = Function(V)
solve(a == L, u, bc)

# Plot solution and mesh
plot(u)
plot(mesh)

# Save solution to file in VTK format
vtkfile = File('poisson2/solution.pvd')
vtkfile << u

# Compute error in L2 norm
error_L2 = errornorm(u_D, u, 'L2')

# Compute maximum error at vertices
vertex_values_u_D = u_D.compute_vertex_values(mesh)
vertex_values_u = u.compute_vertex_values(mesh)
import numpy as np
error_max = np.max(np.abs(vertex_values_u_D - vertex_values_u))

# Print errors
print('error_L2  =', error_L2)
print('error_max =', error_max)

# Hold plot
plt.show()
``````

In poisson It is not showing clearly that error increases in some sense. but we can easily see in case of biharmonic.

``````ravina@ravina:~/Downloads\$ python3 biharmonic.py
Solving linear variational problem.
*** Warning: Degree of exact solution may be inadequate for accurate result in errornorm.
error_L2  = 0.14162139723192527 (with mesh size 1/2)
error_max = 0.3623117469334274
Solving linear variational problem.
*** Warning: Degree of exact solution may be inadequate for accurate result in errornorm.
error_L2  = 0.370765075499436 (with mesh size 1/4)
error_max = 0.6318280069181568
``````

Here is me code file:

``````
from __future__ import print_function
from fenics import *
import matplotlib.pyplot as plt

# Create mesh and define function space
mesh = UnitSquareMesh(4,4)
V = FunctionSpace(mesh, 'Lagrange', 2)

# Define boundary condition
u_D = Expression('sin(pi*x)*sin(pi*x)*sin(pi*x)*sin(pi*x) + cos(pi*x)*cos(pi*x)', degree=2)

def boundary(x, on_boundary):
return on_boundary

bc = DirichletBC(V, u_D, boundary)

# Define variational problem
u = TrialFunction(V)
v = TestFunction(V)
f = Expression('-8*pi*pi*pi*pi*sin(pi*x)*sin(pi*x)*cos(2*pi*x) - 8*pi*pi*pi*pi*sin(pi*x)*sin(pi*x)*cos(2*pi*x) + 8*pi*pi*pi*pi*cos(2*pi*x)*cos(2*pi*x) \
+ 4 *pi*pi*pi*pi*cos(pi*x)*cos(pi*x)', degree=0)
n  = FacetNormal(mesh)
h_avg = (h('+') + h('-'))/2

# Parameters
alpha = 6

# Bilinear form
#print('a=',a)

L = f*v*dx

# Compute solution
u = Function(V)
solve(a == L, u, bc)

# Plot solution and mesh
plot(u)
plot(mesh)

# Save solution to file in VTK format
vtkfile = File('biharmonic/solution.pvd')
vtkfile << u

# Compute error in L2 norm
error_L2 = errornorm(u_D, u, 'L2')

# Compute maximum error at vertices
vertex_values_u_D = u_D.compute_vertex_values(mesh)
vertex_values_u = u.compute_vertex_values(mesh)
import numpy as np
error_max = np.max(np.abs(vertex_values_u_D - vertex_values_u))

# Print errors
print('error_L2  =', error_L2)
print('error_max =', error_max)

# Hold plot
``````