Undergrad Mixed topic : from FEM to analytical solution via limits?

[jk22]

Is this anyhow possible ?
The system would be a wave equation modelized by a finite elements basis in space and time.

Is there any method to do the limit discretization->continuum with paper and pencil ?

 

[S.G. Janssens]

Do you mean a convergence proof of the approximation method?

Rarely will such a proof give you an explicit, analytical expression for the solution, specially not if the domain has a non-trivial geometry. You will need a precise statement about existence and uniqueness of the (weak) solution to the original PDE, and you will need a precise definition of the limit. (These two requirements are usually not independent of each other.)

 

[bigfooted]

There are techniques for computing the analytical solution of your fem discretization. Your analytical solution is then a function of your (constant) mesh size and the polynomial degree of your basis functions. Maybe you can elaborate a bit on what you actually want to achieve.

 

[jk22]

I have never learned well Fem, but I wanted to do spacetime finite elements for the Klein-Gordon equation : $\frac{\partial^2\psi}{\partial x^2}-\frac{\partial^2\psi}{c^2\partial t^2}=\lambda\psi$.

Then I wanted to make the change of coordinates $x'=ct-x,y'=x+ct$ transforming the LHS in $4\frac{\partial^2\psi'}{\partial x'\partial y'}=\lambda\psi'$ ? if I'm not mistaken

Next step was to choose linear basis functions on squares such that their non zero value lie in $x',y'\in[0,2ct]$ to respect the limit speed of $c$. (I think now I should treat this with polar coordinates to be correct)

The usual integration by part to get the stiffness matrix is done and

Strangely it seems to give an eigenvalue problem and the eigenvalue $\lambda=-\frac{m_0c^2}{\hbar^2}$ gives a quantized mass ?

Also there is no initial state to give which seems strange to me.

But I don't know how to code this but rather compute the limit of the steps $\Delta x'=\Delta y'\rightarrow 0$ analytically.

I don't even know if it is a well posed problem nor if it converges.