Integration of (d-1)-dimensional functions on finite element surfaces

Question

I am trying to integrate a function $\hat u$ on the common surface of discontinuous finite elements. The function $\hat u$ lives in a $d-1$-dimensional space of functions defined on the element interfaces. However, the unique mapping of the interface and functions with support on the interface to a reference element is not straightforward.

Consider the interface $\partial K$ joining elements $K^\pm$. We have functions $\hat u$ defined on $\partial K$ and $u^\pm$ defined on $K^\pm$, respectively. It is necessary to compute surface integrals along $\partial K$ from the perspectives of $K^\pm$, i.e. the surface integral is computed twice - once on both element. However, I am having trouble determining the appropriate mapping from $d$-dimensional functions $u^\pm$ to $d-1$ space. Namely, should I enforce a mapping in global coordinates or reference coordinates?

My current approach is to consider a point $x$ on $\partial K$ and to determine the two points $\hat x^\pm$ on reference elements $\hat K^\pm$ that both map to $x$. In this approach, the interface has a fixed global orientation. For example, in the case of the 2-D line interface with a 1-D reference element joining two 2-D triangular elements, the $\hat x^\pm$ mapping to $x$ differ. A consequence of this scheme is that the determinants of the Jacobian are equal in magnitude and opposite in sign on $K^\pm$. However, this scheme seems to not be correct and I'm afraid this approach may be over-complicating the problem.

In particular, I am using integration rules to evaluate integrals of the form

\begin{equation} \int_{\partial K} f_h^\pm \left(\hat u, u^\pm, n^\pm \right) \cdot \mu = \sum_q f \left( \hat u(x_q), u^\pm(x_q), n^\pm \right) \cdot \mu(x_q) J w_q \end{equation}

where $\mu$ is some test function on $\partial K$, $n^\pm$ are the outward-pointing normals to $K^\pm$ and $J$ is the Jacobian of the mapping.

Is my mapping scheme correct? If not, what is the appropriate mapping scheme for, e.g., the the case where $\partial K$ is a line joining two triangles? What would $J$ be for this scheme? Some resources use the absolute value of $J$ in their description of change of variables while others do not. When the absolute value needed?

Answer 1

should I enforce a mapping in global coordinates or reference coordinates?

Yes, it is important to make sure you have a consistent mapping since when you want to compute $f^*_h(u^+, u^-)$ you need to evaluate $u^+$ and $u^-$ on both sides of the jump discontinuity at $x$.

A consequence of this scheme is that the determinants of the Jacobian are equal in magnitude and opposite in sign

This is correct. If you think of this as a flux across the interface, material is moving from one element to the other through the interface. Naturally this means you are subtracting from one side and adding to the other.

What would $J$ be for this scheme?

I don't know the generalization for integrating on a $d-1$ dimensional surface embedded in $d$ dimensional space (I have an idea of what it might look like, but not positive), however I do know what it should be in the special case you have a 1D curve embedded in 2D space or a 2D curve embedded in 3D space.

1D curve in 2D space: $$ \int_{\partial K} \vec{f} \cdot d\vec{S} = \int_{\partial K} \vec{f} \cdot \partial_{\eta} \vec{x} d \eta $$

2D surface in 3D space: $$ \int_{\partial K} \vec{f} \cdot d\vec{S} = \int_{\partial K} \vec{f} \cdot (\partial_{\eta_0} \vec{x} \times \partial_{\eta_1} \vec{x}) d \eta_0 d \eta_1 $$

where $\vec{\eta}$ is the $d-1$ dimensional face coordinates.

It is common to break this further into a non-negative scalar times a unit normal vector, so you get $$ \int_{\partial K} \vec{f} \cdot d\vec{S} = \int_{\partial K} \vec{f} \cdot \hat{n} \|\partial_{\eta} \vec{x}\|_2 d \eta\\ \int_{\partial K} \vec{f} \cdot d\vec{S} = \int_{\partial K} \vec{f} \cdot \hat{n} \|\partial_{\eta_0} \vec{x} \times \partial_{\eta_1} \vec{x}\|_2 d \eta_0 d \eta_1 $$

If your 1D curve is a straight line segment, then you can further simplify $\|\partial_{\eta} \vec{x}\|_2$ to a constant which is the ratio of the line segment's length to the face coordinate's length. ex.: if $\eta \in [-1,1]$, then $$ \|\partial_{\eta} \vec{x}\|_2 = \frac{L_{\partial K}}{2} $$