Homework Assignment 16

The weak form of the steady-state pressure diffusivity equation is

\begin{align} 0 &= \int_{\partial \Omega} \lambda(\vec{x}) \delta p(\vec{x}) \nabla p(\vec{x}) \textrm{d}S - \int_{\Omega} \lambda(\vec{x}) \nabla \delta p(\vec{x}) \cdot \nabla p(\vec{x})\textrm{d}V \ &= \int_{\partial \Omega} \delta p(\vec{x}) \vec{v} \textrm{d}S - \int_{\Omega} \lambda(\vec{x}) \nabla \delta p(\vec{x}) \cdot \nabla p(\vec{x})\textrm{d}V \ &= l(\delta p) - a(p, \delta p) \end{align}

where $\lambda$ is a mobility function, $\delta p$ is a "test function" and $p$ is a "trial function" and unknown pressure field variable. Your assignment is to use Gripap.jl to solve this equation for a one-dimensional discretization. Specifically, you should complete the function fe_solver(number_of_elements::Integer, λ::Function, left_bc::Real, right_bc::Real) in assignmetn16.jl. number_of_elements are the number of finite elements to use in the discretization, λ is the mobility (defined as a function), left_bc and right_bc are the constant left and right Dirchelet boundary conditions respectively. There are no Nuemann boundary conditions, i.e. $\vec{v} = \lambda \nabla p = 0$. The function should return the Gridap solver object (i.e. the thing returned by the solve function).

You should use uniformly defined elements on the one-dimensional domain $(0, 1)$, the reference element should use linear Lagrange interpolation functions, the quadrature should use 2-point Gauss integration.

Testing

To see if your answer is correct, run the following command at the Terminal command line from the repository's root directory

julia --project=. -e "using Pkg; Pkg.test()"

the tests will run and report if passing or failing.