Approximating solutions on non-rectangular domains

[maka89]

Hello!
I have been studying some pertubation theory lately which i found very useful.
I then started thinking about how to approximate solutions to a 2d boundary value problem if the difficulty lies in the geometry of the boundary(I.e. not rectangular), and not in the diff. equation itself(i.e. the diff. equation has a closed form analytical solution for a rectangular domain).

Does anyone have thoughts on how to approximate such solutions if the domain is close to being rectangular?

 

[Geofleur]

If the boundary geometry is close to the "easy" one, you can perturb it. For example, suppose you have a square whose right side, at $x=L$, is a little wonky. You might be able to write your boundary condition as something like $T(x=L+\epsilon(y),y) = 0$. Taylor expand this:

$T(L+\epsilon(y),y) \approx T(L,y) + \frac{\partial T}{\partial x}(L,y)\epsilon(y) = 0$. Now put in your perturbative series, $T = T_0 + \epsilon(y) T_1$ (I've truncated it at first order to keep life simple!). We get

$T_0(L,y)+\epsilon T_1(L,y)+\frac{\partial T_0}{\partial x}(L,y)\epsilon = 0$,

and collecting terms with like powers of $\epsilon$ gives

$T_0(L,y) = 0$,

and

$T_1(L,y) = -\frac{\partial T_0}{\partial x}(L,y)$.

Notice how this mimicks the thing that usually happens: We get the zeroth order approximation and use it to build the first order one, etc. All the boundary conditions are now in terms of what happens at $x = L$. I should note, though, that the $y$ dependence in $\epsilon$ might really complicate the differential equation!

 

[maka89]

Great! thanks for the answer!
But what if the boundary is a little wonky several places and is given for instance as: The value at the curve f(x,y) = 0 should be g(x,y) ? Is there any feasible way of attacking the problem then?

My only hunch is to swith to a curvilinear coordinate system where the boundary is rectangular(I don't know how to do this yet, but am reading up), and treat the terms arising from the scaling factors not being 1 in the differential equation as pertubations.

 

[Geofleur]

Methods of Theoretical Physics, by Morse and Feshbach, has a whole lot about how to the choose coordinate system to fit the problem domain. You might have a look there! I'm not sure what else could be done for complicated cases, other than resorting to numerical methods.

 

[Geofleur]

I forgot to mention, if the solution can be thought of as an analytic complex function, then you can sometimes use conformal mapping methods.

 

[pasmith]

Hello!
I have been studying some pertubation theory lately which i found very useful.
I then started thinking about how to approximate solutions to a 2d boundary value problem if the difficulty lies in the geometry of the boundary(I.e. not rectangular), and not in the diff. equation itself(i.e. the diff. equation has a closed form analytical solution for a rectangular domain).

Does anyone have thoughts on how to approximate such solutions if the domain is close to being rectangular?

If you have an analytical parametrisation of each "side" as a function $f: [0,1] \to \mathbb{R}^2$ then you can construct an analytical map from $[0,1]^2$ to the domain. If the domain is close to rectangular then the map should be close to affine and the Jacobian should be close to constant.