[Insight-users] FEM: PDE of Crank-Nicolson solver

[Cristina Oyarzun]

Hello Brian,

thank you for your answer. I am still confuse about this. In the header file
of the Crank-Nicolson solver (itkFEMSolverCrankNicolson.h) one can find the
next:

" * FEM Solver for time dependent problems; uses Crank-Nicolson implicit
discretization scheme.
 *It solves the following problem:
 *
 *      ( M + alpha*dt* K )*U_t=(M - (1.- alpha)*dt* K)* U_{t-1} +
dt*(alpha*f_{n+1} + (1-alpha)*f_n) "

This problem formulation differs from the one obtained for the heat equation
(for example). So, I guess that it comes from an specific PDE by applying
Crank-Nicolson discretization scheme to it. My questions then are:
1. From which PDE does it come from? Which one is the problem that it is
trying to solve?
2. If the PDE is different depending on the element that one chooses, should
not one get different problem formulations to solve by choosing different
elements (PDE)? And then is it the solver generic or valid for only one PDE?

Thank you!

Cristina

On Fri, Dec 10, 2010 at 8:59 PM, brian avants <stnava at gmail.com> wrote:

> Hi Cristina
>
> The specific PDE depends upon the specific elements you choose.  For
> instance, if you set up a triangular mesh via the
> 2DC0LinearTriangularMembrane  element, then you will use a membrane
> energy penalty term.
>
> The Crank-Nicolson solver uses the Crank-Nicolson approach to
> accumulate the solution to the registration problem over time.
>
> Wikipedia provides a reasonable general explanation of the C-N
> approach and gives an example for the heat equation;
>
> http://en.wikipedia.org/wiki/Crank%E2%80%93Nicolson_method
>
> B.
>
>
> On Fri, Dec 10, 2010 at 10:07 AM, Cristina Oyarzun
> <coyarzunlaura at googlemail.com> wrote:
> > Hello,
> >
> > I am using itk FEM registration filter. This filter makes use of
> > itkFEMSolverCrankNicolson. In the header file of the solver one finds the
> > equation that the solver tries to solve. However, one can only find there
> > its Crank-Nicolson formulation but not the original PDE that leads to
> that
> > equation.
> >
> > I tried to get the discretized equation by using the equation of motion
> that
> > is described in a book and Crank-Nicolson scheme but did not arrive to
> the
> > same solution. Could someone tell me which on is the original PDE that is
> > solved in itkFEMSolverCrankNicolson?
> >
> > Thank you!!
> >
> > Greetings,
> >
> > Cristina
> >
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