2D Laplace equation with additionnal potential : rank form problem

Question

Dear All,

I am trying to reproduce a Onelab (GetDP) problem using Fenics. The problem to solve in the unit square is:
\begin{equation}
\begin{cases}\tag{1}
-\Delta u + u = f & \text{in } \Omega,\
\displaystyle{\frac{\partial u}{\partial \mathbf{n}} = 0} & \text{on }\partial\Omega,
\end{cases}
\end{equation}
with the corresponding weak form:
\begin{equation}\tag{2}
\int_{\Omega} \nabla u\cdot\nabla v \;{\rm d}\Omega + \int_{\Omega}u\, v \;{\rm d}\Omega - \int_{\Omega}fv\;{\rm d}\Omega = 0
\end{equation}
I thought that the corresponding Fenics code would be:



from dolfin import *

import numpy as np

mesh = UnitSquareMesh(64, 64)

mesh.init()

V = FunctionSpace(mesh, "CG", 1)

R = FunctionSpace(mesh, "R", 0)

W = V * R

(u, c) = TrialFunction(W)

(v, d) = TestFunctions(W)

f = Expression("(1+2pipi)cos(pix[0])cos(pix[1])")

a = (dot(grad(u), grad(v)) + cv + du) * dx

L = fvdx - uvdx

w = Function(W)

solve(a == L, w)

(u, c) = w.split()

plot(u, interactive=True)

output to Gmsh .pos file

coor = mesh.coordinates()

u_nodal_values = u.vector()

u_array = u_nodal_values.array()

fp = open('test_laplace.pos', 'w')

fp.write('View "utotm" {\n')

for facet in facets(mesh):

    v0 = facet.entities(0)[0]

    v1 = facet.entities(0)[1]

    v2 = facet.entities(0)[2]

    x0 = coor[v0][0]

    y0 = coor[v0][1]

    z0 = coor[v0][2]

    x1 = coor[v1][0]

    y1 = coor[v1][1]

    z1 = coor[v1][2]

    x2 = coor[v2][0]

    y2 = coor[v2][1]

    z2 = coor[v2][2]

    fp.write("ST(% .13f,% .13f,% .13f,% .13f,% .13f,% .13f,% .13f,% .13f,% .13f){% .13f, % .13f, % .13f };\n"  %(x0, y0, z0, x1, y1, z1, x2, y2, z2, u_array[v0], u_array[v1], u_array[v2] ) )

fp.write('};\n')

print 'Gmsh file test_laplace.pos has been written'

Unfortunately, the second term in (L = fvdx - uvdx) leads to the following error message (last line only):
ufl.log.UFLException: All terms in form must have same rank.

Could someone explain what is wrong in this formulation ?
Thanks.

2D Laplace equation with additionnal potential : rank form problem

1 Answer