Discontinuous Galerkin for Diffusion

  We consider as a model problem the parabolic equation with variable coefficient $\nu$ to demonstrate the treatment of the viscous contributions:

$$u_t = \nabla \cdot (\nu \nabla u) + f, \quad \mbox{in $\Omega$}, \quad u \in L^2(\Omega)$$

$$u = g({\bf x}, t), \quad \mbox{on $\partial \Omega$}$$

We then introduce the flux variable

$${\bf q} = -\nu \nabla u$$

and re-write the parabolic equation

$$u_t = - \nabla \cdot {\bf q} + f, \quad \mbox{in $\Omega$}$$

$$\frac{1}{\nu} {\bf q} = -\nabla u, \quad \mbox{in $\Omega$}$$

$$u = g({\bf x}, t), \quad \mbox{on $\partial \Omega$},$$

The variational formulation of the problem is then as follows: Find $({\bf q},u) \in H({\em div};\Omega)\times L^2(\Omega)$ such that

$$(u_t, w)_{E_i} = ({\bf q}, \nabla w)_{E_i} - <w,{\bf q^B} \c... ..._{\partial E_i} + (f,w)_{E_i}, \quad \forall w \in L^2(\Omega)$$

$$\frac{1}{\nu} ({\bf q}, v) = (u, \nabla v)_{E_i} - <u^B, v {\bf n}\gt _{\partial E_i},\quad \forall v \in H({\em grad};\Omega)$$

$$u = g({\bf x}, t), \quad \mbox{on $\partial \Omega$}$$

where the parentheses denote standard inner product in an element (E_i) and the angle brackets denote boundary terms on each element, with ${\bf n}$ denoting the unit outwards normal and:

The surface terms contain weighted boundary values of v^B, q^B, which can be chosen as the arithmetic mean of values from the two sides of the boundary, i.e. $v^B = \frac{( v^E + v^I)}{2}$, and $q^B = \frac{( q^E + q^I)}{2}$.

Integrating by parts once more, we obtain an equivalent formulation which is easier to implement and it is actually used in the computer code. The new variational problem is

$$(u_t, w)_{E_i} = (-\nabla \cdot {\bf q}, w)_{E_i} - <w,({\bf... ...bf n}\gt _{E_i} + (f,w)_{E_i}, \quad \forall w \in L^2(\Omega)$$

$$\frac{1}{\nu} ({\bf q}, v) = (-\nabla u, v)_{E_i} - <(u^B-u... ..., v\gt _{\partial E_i},\quad \forall v \in H({\em grad};\Omega)$$

$$u = g({\bf x}, t), \quad \mbox{on $\partial \Omega$}$$