Integrals and quadrature used withing a form

Question

Hello,

I would like to define a function that includes an integral of an unknown in a single direction of a 3D mesh, like this:

$$ U(x,y,z) = \int_{z_b}^z u(x,y,z') dz' $$

I would like an expression for the continuous definite integral of $u(x,y,z)$. For example, using rectangular quadrature,

$$ U(x,y,z) = \int_{z_b}^z u(x,y,z') dz' \approx (z-z_b) u\left(x,y,\frac{z_b+z}{2}\right).$$

I keep seeing "quadrature" elements -- can they be used to solve this problem? I'd rather use built-in tools if I can, but otherwise I'll have to use numpy or something.

Comments

Hello pf4d, your notation is ambiguous to me. One thing is that you're integrating over z' but use z both in the integrand and as upper limit. Another is that the left hand side should still depend on x,y? Using different letters, do you mean this?

f(x,y) = \int_a^b u(x,y,z) dz

But sorry, I don't see how to solve this using built-in tools, and no, quadrature elements is something entirely different.

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Yes, I think this may have been ambiguous -- I've edited the equation to better describe what I'm doing.

If I were working on a 2D mesh, then I could just define several $u$ functions in sigma coordinates and use them for the depth-integrated values. However, my problem is defined on a 3D mesh, and as such I cannot separate the layers in a UFL form explicitly, like $u(0.5, 0.5, 0.5)$.

My best solution to this problem is create my own Newton solver and at each iteration solve the integration problem explicitly, as described here.

I thought that I could maybe hack the quadrature parts of FEniCS used for FEM machinery to do this in a neater way.