[petsc-users] Solving/creating SPD systems

[Jed Brown]

Justin Chang <jychang48 at gmail.com> writes:

> So I am wanting to compare the performance of various FEM discretization
> with their respective "best" possible solver/pre conditioner. There
> are saddle-point systems which HDiv formulations like RT0 work, but then
> there are others like LSFEM that are naturally SPD and so the CG solver can
> be used (though finding a good preconditioner is still an open problem).

LSFEM ensures an SPD system because it's effectively solving the normal
equations.  You can use CGNE if you want, but it's not likely to work
well.  Note that LS methods give up local conservation, which was the
reason to choose a H(div) formulation in the first place.  You can use
compatible spaces/Lagrange multiplies with LS techniques to maintain
conservation, but you'll return to a saddle point problem (with more
degrees of freedom).  There's no free lunch.

> I have read and learned that the advantage of LSFEM is that it will always
> give you an SPD system, even for non-linear problems (because what you do
> is linearize the problem first and then minimize/take the Gateaux
> derivative to get the weak form). But after talking to some people and
> reading some stuff online, it seems one could also make non SPD systems SPD
> (hence eliminating what may be the only advantage of LSFEM).

(Symmetric) saddle point problems have an SPD Schur complement.  Schur
complements are dense in general, but some discretization choices (e.g.,
quadrature in BDM) can make the primal block diagonal or block-diagonal,
resulting in a sparse Schur complement.  If the Schur complement is
dense, you might be able to approximate it by a sparse matrix.  The
quality of such an approximation depends on the physics and the
discretization.
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