Rajat,

The geometry is a cubical box [0 1] X [0 1] X [0 1] with a uniform rectilinear mesh in the domain.

I have obtained the scalar solution vector u in the domain.

I need to find the line integral of the solution along the boundary. For example: lets say
I want to integrate the solution variable
\int u dx on the line which is on +z face and goes from from (0,0,1) to (1,0,1).

There is not functionality right now that allows you to do easily what you want to do. At some fundamental level, the reason is that for solutions of most PDEs, the solution is in the space H^1 which allows you to take integrals over d-dimensional objects (e.g., cells) as well as (d-1)-dimensional objects (e.g. faces) but not (d-2)-dimensional objects such as lines in 3d. It's just not well defined mathematically.

On a pragmatic level, it's not available because nobody has implemented this.


Can someone please help to figure out how to do this? In the pseudo code, my doubt is how to initialize the fe_face_values (line values?) object and the gauss quadrature formula for this?
You have to create a 2-dimensional quadrature object that you can then use in FEFaceValues. To this end, you need the 2d quadrature objects to have only quadrature points on one of the bounding lines. This can be done using the QProjector class. The issue you have to pay attention to is *which* edge of the face quad you want to integrate over. This may require a bit of book-keeping.

Best
 W.

--
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Wolfgang Bangerth          email:                 bange...@colostate.edu
                           www: http://www.math.colostate.edu/~bangerth/

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