🚀 go-pugleaf

In article <ROGER.93Jan27083432@kea.grace.cri.nz>, roger@maths.grace.cri.nz (Roger Young) writes:
|> 
|> I have been following the correspondence in the num-analysis
|> newsgroup concerning FE methods as applied to symmetric but
|> non-positive definite matrices.
|> 
|> I appear to have a similar problem with the poro-elastic 
|> equations. These equations are: (1) the elastic equilibrium
|> equations with the pressure gradient as a source term (or,
|> in fact, it can be convenient to work with the variation
|> of these equations)
|> 
|> 	grad(j)d(stress)(ij) = grad(i)d(pressure)
|> 
|> (2) the pressure diffusion equation with an additional time
|> derivative de/dt of the elastic dilatation e = 
|> grad(i)(displacement)(i)
|> 
|> 	S.d(pressure)/dt + de/dt = -grad(i)q(i)
|> 
|> where the storage S can be taken as constant, and q is the
|> Darcy velocity which is proportional to pressure gradient
|> q(i) prop grad(i)(pressure).
|> 
|> In the displacement-pressure formulation the field vector
|> is chosen as z = (displacement, pressure)^T. Then according
|> to Lewis and Roberts "The Finite Element Method in Porous 
|> Media Flow" (in Fundamentals of Transport Phenomena in Porous
|> Media, eds Bear and Corapcioglu), the finite element equations
|> for the coupled problem can be written in the form
|> 
|> 	K(dz/dt) + Hz = f
|> 
|> where K is SYMMETRIC.
|> 
|> If elasticity is decoupled from pressure, then it appears that 
|> the reduced stiffness matrix is positive definite for displacement
|> and pressure individually. However, for the coupled problem
|> numerical analysis appears to show that K has both +ve and -ve
|> eigenvalues, ie is not positive definite.
|> 
|> My question(s) are: is this really true? How does one set about
|> proving (or disproving) positive-definiteness? what is the
|> physical significance of positive-definiteness?
|> 
|> Any help from readers would be appreciated...


This at first glance appears to akin to the problem  that crops up with Lagrange
multipliers (which can masquerade as pressures under many conditions).  If this
is true, then solving for the all elasticity terms connected with a given node
before you solve for any pressure terms for a given node you should find typical
factorization algorithms to work (dont try cholesky decomposition however).

Postive definitenes is in my experience as muchg a religious phenomena as
anything else (opps here go the battles).  Consider however that  if I multply
both sides of a positive definite system by -1 I get a negative definite system,
which is still very well behaved.  In  my experience and from  my religious
framework positive definite means that I have a positive energy functional which
can be minimized (negative definiteness means to me that I have a negative enrgy
functional which needs to be maximized!).

Problems seem to occur for the following conditions:
 
     indefinite - zero eigenvalues
     semidefinite - some are postiive, some are negative

which seems to be your case.  In some cases a semidefinite system can  be
organized as a positive definite system followed by a negative definite system.

Hope these philosophical ramblings more or less answered your question.
-- 

Regards.
 
Don Rolph  a722756@pan.mc.ti.com WD3 MS10-13 (508)-699-1263

>My question(s) are: is this really true? How does one set about
>proving (or disproving) positive-definiteness? what is the
>physical significance of positive-definiteness?

The existence of a definite coefficient matrix is a sign that the 
operator form of the problem can be obtained by an extremum principle
(minimum principle => positive definite, max => neg def). This is
typically possible for self-adjoint, (positive) definite operators. 
In this case, the Ritz method and the Galerkin method are equivalent, 
and it can be shown that the resulting approximation is optimal, in the 
sense that it minimizes the discretization error (in an "energy" norm). 
It is easy to show that the coefficient matrix must be (positive) 
definite. The classic example is linear elasticity; the minimum 
principle is minimum potential energy or minimum complimentary energy. 
For example, the differential equation for axial deformation of an 
elastic bar -ku" = f can be obtained by finding u(x) that minimizes 
the functional \int 0.5k(u')^2 - fu dx. On the other hand, the 
convection-diffusion equation -ku" + cu' = f is not derivable by 
extremizing a functional, so we cannot expect the coefficient matrix 
be definite. 

Numerically, many nice properties accrue from positive definiteness. 
With direct methods, we need not worry about stability and are free to 
pivot to reduce fill-in. Certain iterative methods (e.g. Jacobi, Gauss 
Seidel, conjugate gradients) are guaranteed to converge from any 
starting iterate.