# Integrate on a specific element

Hi,

I am trying to integrate the product of two Gridfunction over a specific element. I know we can achieve this by setting “element_wise=True” as in " Integrate(gf1*gf2,mesh,element_wise=True)".

However, this easy option computes the integral value over all the elements and returns a vector made of all these values. This doesn’t seem to be very optimal in computational efficiency if I only need the integral value over one specific element.

I am wondering if there is any other convenient way to compute such an integral over only one element using NGSolve.

Many thanks in advance for any suggestion.

Best regards.

Hi,

you can use a BitArray to mark that one element and integrate using the definedonelements flag. The code would look something like

``````marker = BitArray(mesh.ne)  # mesh.ne = number of elements
marker.Clear()
marker.Set(i) # with i the number of the element
Integrate(gf1 * gf2 * dx(defineonelements=marker), mesh)
``````

As far as I know, this is an efficient way to only integrate over the marked elements.

Best wishes,
Paul

Hi Paul,

Thanks a lot for your help. I think using “Integrate(gf1 * gf2 * dx(defineonelements=marker), mesh)” doesn’t work for me, while the symbolic integrator/bilinear forms works well as follows.

``````mesh = Mesh(unit_square.GenerateMesh(maxh=0.5))
marker = BitArray(mesh.ne)  # mesh.ne = number of elements
marker.Clear()
marker.Set(4) # with i the number of the element
U = H1(mesh, order=1)
gf = GridFunction(U)
gf.Set(1)
a = BilinearForm(U)
a += U.TrialFunction()*U.TestFunction() * dx(definedonelements=marker)
print(2*a.Energy(gf.vec))

``````

Moreover, I am wondering if there is a similar convenient way that we can integrate over a specific edge by any chance. Is there something like ds(…) specifying the integrator on one edge? Many thanks again.

Best regards.

Strange, for me `Integrate(gf1 * gf2 * dx(defineonelements=marker), mesh)` works totally fine. You can add `element_boundary=True` inside the DifferntialSymbol (dx). Then, you integrate over the boundaries of that element (i.e. the facets). I don’t know how to integrate only over one specific facet using a marker (ds integrates over the boundary of the mesh, not the given elements). But you can define a facet indicator function via `GridFunction(FESpace("facet", mesh, order=0))` and set the values on all facets to zero besides the ones you want to integrate on. Multiping this indicator with your function and only integrating over the edges of the specific elements should lead to the desired output.

Best whishes,
Paul

Hi Paul,

Strange, for me `Integrate(gf1 * gf2 * dx(defineonelements=marker), mesh)` works totally fine.

Could the reason be that we are using different versions of NGSolve? I am using NGSolve-6.2.2304…

Best regards.

Hi,

that is possible (I am not an expert either). I don’t know the version I use by heart, but it is the latest version from the master branch from git.

Hi,
it’s not a single Python statement, but you can use an integration rule for that:

``````from ngsolve import *
mesh = Mesh(unit_square.GenerateMesh(maxh=0.3))

ir = IntegrationRule(TRIG,5)
cf = x

globigl = 0
for el in mesh.Elements(VOL):
trafo = mesh.GetTrafo(el)
igt = sum( ip.weight*trafo(ip).measure*cf(trafo(ip)) for ip in ir )
globigl += igt
print (igt)
print ("global integral", globigl)
``````

Another idea is to compute the element-vector for a finite element element with one shape-function constant 1 (i.e. an L2(order=0) -finite element):

``````fes = L2(mesh, order=0)
v = fes.TestFunction()
lfi = (cf*v*dx)[0].MakeLFI()

globigl = 0
for el in mesh.Elements(VOL):
trafo = mesh.GetTrafo(el)
fel = fes.GetFE(el)
igt = lfi.CalcElementVector(fel, trafo)[0]
globigl += igt
print (igt)
print ("global integral", globigl)
``````

Joachim

Hi Joachim,

Got it. Thanks a lot for your help.

you can use an integration rule for that

I am also wondering if there is any convenient way to set up the quadrature rule on edges so that we can compute the integral over one specific edge. Is it possible to have an analogue of your first solution in the case of integrals on edges? Or should I get the coordinates of points on the edge and set the quadrature rule manually?

Many thanks in advance for any suggestion.

Best wishes.