- Of particular importance to our simulation is the Darcy Velocity:

$$$$\vec{u} = -\frac{\mathbf{K}}{\mu} \nabla p$$$$

- The Darcy velocity appears in the heat convection-conduction equation:

$$$$C\left( \frac{\partial T}{\partial t} + \epsilon \vec{u}\cdot\nabla T \right) - \nabla \cdot k \nabla T = 0$$$$

- The primary unknown (aka nonlinear variable) is the pressure.
- Once the pressure is computed, the
`AuxiliarySystem`

can compute the velocity field and write it to the output file. - We must couple in the primary nonlinear variable's gradient and make use of
`Material`

properties in the auxiliary computation. - Auxiliary variables come in two flavors: Nodal and Elemental
- Nodal Auxiliary variables cannot couple to gradients of nonlinear variables.
- Gradients of $$$C^0$$$ continuous nonlinear variables are not well-defined at the nodes.

- Elemental Auxiliary variables (i.e.
`family = MONOMIAL`

) can couple to gradients of nonlinear variables -- they are only evaluated in element interiors.

- Build a test for
`DarcyVelocity`

. - To simplify the test, we:
- Remove Kernels and disable the solve.
- Create an artificial pressure field using
`FunctionIC`

.

tutorials/darcy_thermo_mech/step04_velocity_aux/tests/auxkernels/velocity_aux/tests

tutorials/darcy_thermo_mech/step04_velocity_aux/tests/auxkernels/velocity_aux/velocity_aux.i