🚀 go-pugleaf

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...

Roger Young (roger@maths.grace.cri.nz)