Hi, I want to impose Dirichlet boundary conditions by using the Lagrange multiplier method.
The governing equation for heat transfer is \Delta T = 0, where T is the temperature. The boundary conditions are shown in the following picture.
I know how to solve this problem in FreeFEM. However, I try to impose constant temperature on the top and bottom boundaries by using a Lagrange multiplier method, and the weak form is shown below:
\Biggl\{ \begin{array} -\int \nabla T \cdot \nabla w \; \text{d} \Gamma+\int w \mathbf{n} \cdot \nabla T \text{d} \Gamma + \int \lambda w \; \text{d} \Gamma &= 0, \\ \int \hat{\lambda} \left( T - T_c \right) \; \text{d} \Gamma &= 0, \end{array}
where w and \hat{\lambda} is the test function for temperature T and Lagrange multiplier \lambda. Note that here \lambda is a scalar field defined on the top and bottom boundaries.
The FreeFEM provides a Lagrange multiplier example (Poisson equation with a global constraint). In this example, we only need to introduce a scalar to enforce the global constraint, which is different from our case.
Does FreeFEM have a similar case that enforces boundary constraints by introducing a Lagrange multiplier?
Can FreeFEM define a finite element space for the Lagrange multiplier defined on the specific boundary ?
Thanks in advance.
