dolfin team mailing list archive

[Anders Logg <logg@xxxxxxxxx>]

On Fri, Sep 23, 2005 at 05:07:58PM +0200, Johan Jansson wrote:

> > To have FFC work with the ODE solver is one thing. What I meant is to have
> > FFC automatically discretize a PDE in space-time, that is we do not write
> > out the discretization of the derivatives in time for example ((u-u0)/k)
> > but instead we write u.dt().
> > 
> > The heat equation would then be something like:
> > 
> > a = v.dx(i)*u*dx(i)*dx*dt
> > f = v*f*dx*dt

This might ultimately be the way to do it, since that allows the most
flexibility for a user to define the space-time discretization.

> I think a nice solution for this would simply to write all PDEs in the form:
> 
> dot(u) = f(u, t)
> 
> (which is what Anders is talking about in his dissertation)
> 
> The heat equation:
> 
> dot(u) - laplace(u) = g(t)
> 
> would be:
> 
> integral(dot(u)*v) = integral(grad(u)*grad(v) + g(t)*v)
> 
> then we would need something to handle the mass matrix integral(u*v)
> on the left hand side. An initial version could perhaps simply do
> lumping.

This is probably the easiest thing to do right now. The ODE solver
(the monoapaptive version) can solve Mu' = f(u) so we wouldn't need to
lump the mass matrix.

> This would mean that the space discretization is handled by f() and
> the time discretization by the ODE solver which is fed the dot(u) =
> f(u, t) problem. I suppose there would need to be an extra interface
> between the discretizations for mesh refinement, but hopefully it
> won't be too complicated. Any comments on this from someone who has
> done mesh refinement for time-dependent problems?
> 
> For non-linear problems, f(u, t) needs to be recomputed anyway. And
> computing the action f(u, t) is cheaper than to first compute the
> matrix, then compute the matrix-vector product.
> 
> For linear problems, an optimization could be made by pre-computing
> the Jacobian df/du. The standard right hand side of a PDE could be
> computed by f(0, t). Perhaps f(u, t) could be split up into two
> different parts, one dependent on u and one not, to simplify this type
> of optimization. The overhead of this approach compared to explicitly
> assembling the matrix and vector part should be very small (just a
> small startup cost).
> 
> For time-independent problems, we have dot(u) = 0. So the ODE solver
> could have a special case for that, and simply solve 0 = f(u, 0).
> 
> What do you think? It would be even nicer if the solver would be able
> to detect if the problem was non-linear or linear, or time-dependent
> or not, automatically, and select the most optimal solution
> strategy. But initially perhaps this information can be supplied with
> the PDE.

There can be some intelligence built into the PDE class (should we
call it PDE, LinearProblem, VariationalProblem or what?) to detect
nonlinearity, linearity, terms that are constant etc.

/Anders