Integral Transform?

Hello! I’m interested in using an integral transform within a problem I’m solving, and I don’t see an obvious way forward.

In general, let’s say I need an operator “A” that sends a function u (in a FE space Vh) to another space (Uh). For now, let the two spaces Vh and Uh use the same basic rectangular domain, on the same mesh. (Adaptive meshing in the Vh space is the goal, but we’ll cross that bridge when we get to it.)

I’d like “A” to be the Abel transform eventually, but for the sake of simplicity, assume I’m after something like this:

g(x,y) = int_0^x f(\tau, y) d\tau

Is there a straightforward way to accomplish this?

Remark, the function g is not in finite element space so you interpole in space Uh.

This operator build a full matrix type, it is possible to build directly the full but it is very expensive.

No more idea.