20. Parabolic equations

20.1. Heat equation

Heat equation for a scalar field \(\phi(\vec{r},t)\) can be interpreted as the unsteady equation of a Poisson equation,

\[\partial_t \phi - \nabla \cdot (\nu \nabla \phi) = f \qquad (\vec{r}, t) \in V \times [0, T] \ ,\]

with proper boundary and initial conditions, \(\phi(\vec{r},0) = \phi_0(\vec{r})\). Common boundary conditions are the same as the one discussed for Poisson problem.

20.1.1. Weak formulation

For \(\forall w \in \dots\) (functional space, recall some results about existence and uniqueness of the solution, Lax-Milgram theorem,…)

\[\begin{split}\begin{aligned} 0 & = \int_V w \, \left\{ - \partial_t \phi + \nabla \cdot (\nu \nabla \phi) + f \right\} = \\ & = \oint_{\partial V} w \hat{n} \cdot (\nu \nabla \phi) + \int_V \left\{ - \partial_t \phi - \nu \nabla \vec{w} \cdot \nabla \phi + w f \right\} = \\ \end{aligned}\end{split}\]

Splitting boundary contribution as the sum from single contributions from different regions, and applying boundary conditions, setting \(w = 0\) for \(\vec{r} \in S_D\) (see the ways to prescribe essential boundary conditions),

\[\begin{aligned} 0 = \int_{S_D} \underbrace{w}_{= 0} \hat{n} \cdot (\nu \nabla \phi) + \int_{S_N} w \underbrace{\hat{n} \cdot (\nu \nabla \phi)}_{ = h} + \int_{S_R} w \underbrace{ \hat{n} \cdot (\nu \nabla \phi)}_{ = k - \phi } + \int_V \left\{ - \partial_t \phi - \nu \nabla \vec{w} \cdot \nabla \phi + w f \right\} \ . \end{aligned}\]

and rearranging the equation separating terms containing unknowns from known contributions,

\[\int_{V} w \partial_t \phi + \int_{V} \nu \nabla w \cdot \nabla \phi + \int_{S_R} w \phi = \int_{V} w f + \int_{S_N} w h + \int_{S_R} w k \qquad \forall w \in \dots \ ,\]

and \(\phi = g\), for \(\vec{r} \in S_D\).