March 1, 2002
In this project we shall investigate the behaviour of multigrid on a heat or diffusion equation. We suggest that the Heat1 (or Heat2 ) simulator in the
Diffpack distribution is extended with the MGtools .
The Heat1 simulator solves the problem u ( x , t ) = g ( x , t ) , x ∈ ∂ Ω
E
∂u
= ∇ · ( k ∇ u ) + f, x ∈ Ω , t > 0 ,
∂t
, t > 0 , u ( x , 0) = I ( x ) , x ∈ Ω , t = 0 .
A finite element method is used to discretize in space, while a backward finite difference is used in time.
The first task is to extend the simulator with MGtools such that multigrid methods can be used. The details of how this extension should be done is outlined in [1] and the associated foils. Notice that multigrid should be used to solve the linear system on each time step.
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A number of numerical experiments should be carried out.
1. What kind of convergence criterion is relevant is these experiments?
2. Does it make sense to set up a ”proper” test example like the random start vector in this problem?
3. How does multigrid behave in terms of dt and h ?
4. Does multigrid or nested multigrid reach discretization error in O ( n ) iterations ?
5. How does multigrid compare with a Krylov solver with an algebraic preconditioner like, e.g. RILU?
6. How does multigrid as a stand alone solver compare with Krylov solvers that utilize multigrid as a preconditioner?
The numerical experiments should be presented in tables and convergence rates should be calculated.
[1] Multigrid Methods in Diffpack, Kent-Andre Mardal, Gerhard W. Zumbusch, Hans Petter Langtangen.
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