What is the best approach to go about solving a PDE problem of the type

\begin{equation} k^3\Delta u - k(\mathbf{1}\cdot\nabla u) = 0\, ,\\ u=g\; \text{on}\; \Gamma_D\, ,\\ mean(u) = u_\text{mean} \end{equation}

where one wants to find for which positive constant scalar coefficient $k>0$ the mean of the solution $u$ fulfills a prescribed value $u_\text{mean}$?

  • 1
    $\begingroup$ I assume $k$ is a vector to be dimensionally correct for $k \nabla u$ and $k^3=|k|^3$, is the direction of $k$ known? Otherwise you have to few conditions. If so, you can add your condition $mean(u)-u_\text{mean}=0$ with a lagrange multiplier to your equation and solve the system for $|k|$ as free parameter. $\endgroup$ – Bort Nov 13 '18 at 17:27
  • $\begingroup$ Ah no, k is just a constant scalar in this case. $\endgroup$ – jjparm85 Nov 14 '18 at 3:00
  • $\begingroup$ if $k$ is a constant scalar, then the dimensions don't match up and writing $k^3\Delta u + k\nabla u$ doesn't make much mathematical sense... is there a unit vector field hiding somewhere in there i.e. do you mean to write $\mathbf{1} \cdot \nabla u$? $\endgroup$ – GoHokies Nov 14 '18 at 14:45
  • $\begingroup$ Yes, that's right a unit vector for the second term, thanks. $\endgroup$ – jjparm85 Nov 16 '18 at 11:07

I would recommend an adjoint method also but slightly different. Following is the general idea. Whether this approach is well posed for this problem is left as an excercise :-)

Solve this $$ \min_k J(k) = (1/2)[u_{mean} - mean(u)]^2 $$ subject to the pde as constraint. If you change $k$ to $k+k'$, $u$ changes to $u+u'$ and $J$ to $J+J'$. Find a linear PDE for $u'$ assuming changes are small, which should have following structure $$ L(u',k)=F(u,k) k' $$ and it will have $u'=0$ on boundary since you have a Dirichlet bc on $u$. Then you get $$ J' = -[u_{mean} - mean(u)] mean(u') + \int_\Omega v \cdot L(u',k) dx - \int_\Omega F(u,k) k' v dx $$ and we just added a zero term. Integrate by parts so all derivatives are on the adjoint variable $v$. Then you get an equation of the form $$ J' = - k' \int_\Omega F(u,k) v dx + \int_\Omega L^*(v,k)\cdot u' dx + \textrm{boundary terms} $$ Form adjoint pde $$ L^*(v,k)=0 $$ and choose bc for $v$ so that boundary terms in $J'$ vanish. Once you solve adjoint, you can update $k$ as $$ k' = \int_\Omega F(u,k) v dx $$ which ensures that $J$ will decrease.


With $$k^3\Delta u-k(1\cdot\nabla u)=0,$$ you can already remove 1 solution for $k$, $k=0$ which is not of interest for you. You are really looking at $$k^2\Delta u-(1\cdot\nabla u)=0.$$

Edit: The previous equations enforced for a given $k$ and $u_\text{mean}$ a solution on the field $u$ and its constraint force $\lambda$.

If you want to find $k$ for a given $u_\text{mean}$, it should be rephrased to an optimization under constraints: $$\min_{k>0} \frac{1}{2}||PDE(k,u)||^2+\frac{\lambda}{2}^2||u_\text{mean}-\text{mean}(u)||^2$$

Here $\lambda$ is a regularization parameter, basically how much weight you put on the constraint. In this case you end up with a similar equation:

$$k^2\Delta u-(1\cdot\nabla u)+\lambda(u_\text{mean}-\text{mean}(u))=0.$$ In this optimization problem you are no longer solving for $\lambda$ (this is input) but for $k$.

In case of optimization I suggest to look for the adjoint formulation of PDEs with constraints, e.g. the mathematical description on the dolfin adjoint page and its references.

  • $\begingroup$ Using this technique, how do you find the $k$ which naturally satisfies the condition $\text{mean}(u) = u_\text{mean}$? To me it is quite obvious that such $k$ exists. This technique basically adds an artificial force field $\lambda$ which causes the constraint $\text{mean}(u) = u_\text{mean}$ to hold but then the resulting $u$ does not satisfy the original equation anymore. $\endgroup$ – knl Nov 20 '18 at 12:35

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