Weak Form of the Poisson Problem

[Morberticus]

Hi,

I know the weak form of the Poisson problem

$\nabla^2 \phi = -f$

looks like

$\int \nabla \phi \cdot \nabla v = \int f v$

for all suitable $v$. Is there a similarly well-known form for the slightly more complicated poisson problem?

$\nabla (\psi \nabla \phi ) = -f$

I am writing some finite element code and variational/weak forms are very handy.

Thanks in advance

 

[pasmith]

Hi,

I know the weak form of the Poisson problem

$\nabla^2 \phi = -f$

looks like

$\int \nabla \phi \cdot \nabla v = \int f v$

for all suitable $v$. Is there a similarly well-known form for the slightly more complicated poisson problem?

$\nabla (\psi \nabla \phi ) = -f$

I am writing some finite element code and variational/weak forms are very handy.

Thanks in advance

By the product rule
$$\int_V v\nabla \cdot(\psi \nabla \phi)\,dV = \int_V\nabla\cdot(v\psi \nabla \phi) - \psi (\nabla \phi) \cdot (\nabla v)\,dV = \int_{\partial V} v\psi \nabla \phi \cdot dS - \int_V\psi (\nabla \phi) \cdot (\nabla v)\,dV$$
and hence the weak form of $\nabla \cdot(\psi\nabla\phi) = - f$ is
$$\int_V \psi (\nabla \phi) \cdot (\nabla v)\,dV = \int_V fv\,dV$$