# DG local equation, how to interpret mean-averaged test function

In the paper http://www.sciencedirect.com/science/article/pii/S0045782509003521, an HDG element-local equation is described on page 584 equation (4), with one of the equations taking the following form

$$-(u_h,\nabla q)_K = -\left\langle\hat{u}_h \cdot n, q - \bar{q}\right\rangle_{\partial K}$$

Which is the variational approximation to the continuous equation $\nabla \cdot u = 0$, with a scalar-valued test function $q$ in a space that makes sense.

The paper defines $$\bar{q} = \frac{1}{|\partial K|} \int_{\partial K} q$$.

How is this interpreted, in a finite element sense? From my understanding, we multiply both sides by a test function $q$ and then attempt to find the solution which satisfies the equation for all possible choices of $q$. How is it possible to modify the test space in this manner?

The paper also states that this is necessary to enforce the identity $$\left\langle\hat{u}_h\cdot n, q - \bar{q}\right\rangle_{\partial K} = 0$$ I agree with this statement, but how might a test function $q - \bar{q}$ be implemented in code? Should I take the basis functions on the element and subtract their mean when assembling the element local linear system?

• Have you tried contacting the authors of the paper themselves? – Paul Jul 2 '17 at 3:52

I suppose that the space function in which $q$ is sought has a null mean, i.e. the new test functions $q^*$ have been defined such as: $$\int_{\Omega}{q^*\,dx} = \int_{\Omega}{(q-\overline{q})\,dx} = 0$$ This is common in systems in which a compatibility condition arises must be fulfilled.