How to redefine jump operator for a space-time problem?

Question

Hi everyone! I'm new to FEniCS and would like to solve a PDE of the form $\partial_{tt} u - \Delta^2 u = 0$ using a space-time Galerkin method, i.e., I would like to treat the time like a space dimension.

I see in FEniCS that there is C0 interior penalty method to implement $\Delta^2 u = 0$. However in my case, I have to redefine the normal components used in the jump operator. I have written the following code for a 1D+1D (space+time) problem:

def myjump(v, n): "UFL operator: Take the jump of *v* across a facet." v = as_ufl(v) r = len(v.ufl_shape) if r == 0: warning("not defined: myjump") else: return v('+')[0]*n('+')[0] + v('-')[0]*n('-')[0]

My question is how can I scale the following expression?

v('+')[0]*n('+')[0] + v('-')[0]*n('-')[0]

I cannot simply divide by n[0] as it might divide the expression by zero, and I'm not also sure if division operation is defined with such data structure.

Answer 1

Hi, how about conditional

from dolfin import *

mesh = UnitSquareMesh(10, 10)
V = VectorFunctionSpace(mesh, 'CG', 1)
v = interpolate(Expression(('x[0]', '2*x[1]'), degree=1), V)
n = FacetNormal(mesh)

def myjump(v, n, tol=1E-12):
    j = v('+')[0]*n('+')[0] + v('-')[0]*n('-')[0]
    return conditional(abs(n('-')[0]) > tol, j/n('-')[0], j)

print assemble(myjump(v, n)*dS) 

Comments

Thanks! Do you know perhaps why it crashes with optimization flags?
I used the following two:

parameters["form_compiler"]["cpp_optimize"] = True

and

parameters["form_compiler"]["optimize"] = True

---

Hi, looking at the error.log I see something like

/tmp/tmpk6qs8B/ffc_form_e9a89afcd2a6fe79628a1f5625f7eab29646345e.cpp:170:32: error: ‘n_00’ was not declared in this scope
           C[0] = ((std::abs( - n_00) > 1e-12)) ? (F0*n_00 - F1*n_00)/( - n_00) : (F0*n_00 - F1*n_00);

so wrong code was generated. This could be a bug in FFC.