"maximum error at vertices" increased a lot when use irregular mesh instead of regular mesh

General
1

Hello everyone. I learned the Fenics tutorial and then tested the ft01_poisson.py provided by the Fenics tutorial. When I used a regular mesh with 81 nodes, the “maximum error at vertices” was “1.3322e-15”. However, when I used an irregular mesh with 90 nodes, the “maximum error at vertices” became “0.005291”. The “maximum error at vertices” increased a lot, which doesn’t make sense because the irregular mesh and regular mesh I used have very close numbers of nodes (one has 81 nodes and the other has 90 nodes).
Any help will be greatly appreciated. Thank you very much in advance.
I presented the code below, which is just the ft01_poisson.py provided by Fenics tutorial.

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

# the regular mesh with 81 nodes:
mesh = UnitSquareMesh(8, 8)

# the irregular mesh with 90 nodes:
# domain = Rectangle(Point(0,0), Point(1,1))
# mesh = generate_mesh(domain, 7)

V = FunctionSpace(mesh, 'P', 1)

num_dofs = V.dofmap().global_dimension()
print("num_dofs:",num_dofs)

# Define boundary condition
u_D = Expression('1 + x[0]*x[0] + 2*x[1]*x[1]', 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)
a = dot(grad(u), grad(v))*dx
L = f*v*dx

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

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

# 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()
2

Try changing 1 into 2. Since the exact solution is a polynomial of degree 2, using a P2 finite element space will give you the exact solution up to machine precision.

In general, if you only use a P1 FE space you will get non zero error: in that case, you may want to iteratively decrease the mesh size and verify that the convergence rate is the expected one.

3

Dear Ballarin,

It works. Using the P2 finite element, the “maximum error at vertices” decreased to “9.3036e-14”. However, the Fenics tutorial said “The machine precision is about 1e-16”. Can we think that “9.3036e-14” is within the machine precision? Thank you very much for your kind help.

Best regards

4

Yes, you can. Keep in mind that there are operations (solving the linear system, computing the error) that may introduce some additional small errors on top of the machine precision.

5

Dear Ballarin,

Got it. Thank you very much for your kind help.

Best regards