FEniCS 2019.1 tutorial 1

The first FEniCS tutorial, modelling the Poisson equation, is computed to make sure everything works.

FEniCS tutorial demo program: Poisson equation with Dirichlet conditions

Test problem is chosen to give an exact solution at all nodes of the mesh.

inline_formula not implemented in the unit square

inline_formula not implemented on the boundary

inline_formula not implemented

inline_formula not implemented

Import packages.

from fenics import *
1.1s
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Create mesh and define function space

mesh = UnitSquareMesh(8, 8)
V = FunctionSpace(mesh, 'P', 1)
1.4s
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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)
2.4s
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Define variational problem

u = TrialFunction(V)
v = TestFunction(V)
f = Constant(-6.0)
a = dot(grad(u), grad(v))*dx
L = f*v*dx
0.0s
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Compute solution.

u = Function(V)
solve(a == L, u, bc)
4.1s
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Plot solution and mesh.

plot(u)
0.5s
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plot(mesh)
0.4s
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Save solution to file in VTK format.

vtkfile = File('results/solution.pvd')
vtkfile << u
0.0s
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Compute error in L2 norm.

error_L2 = errornorm(u_D, u, 'L2')
11.5s
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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))
0.0s
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Print errors

print('error_L2  =', error_L2)
print('error_max =', error_max)
0.3s
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0.0s
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Runtimes (1)