# Is it possible to predict the null space of a structure from contributing elements null spaces?

I am trying to solve an almost incompressible problem with heterogeneous properties by domain decomposition. Solution with CG converges slowly or divergerces completely. My problem becomes ill-conditioned (condition number in the order of 10^5) due to the poissons ratio approaching to 0.5 and stiffness differences in between materials. I detected that in element stiffness matrix, several additional zero energy modes (hourglass modes etc) are formed in addition to standard rigid body modes.

I am planning to prevent these modes by MatNullSpace object in Petsc. Since the computation of nullspaces with SVD or QR is expensive, I am trying to find a cheaper way of predicting the nullspace of floating subdomains.

Is it possible to predict nullspaces for subdomain from contributing finite element's null spaces?

I read a broad sentence in a publication says that "The rigid body modes of a collection of elements is equal to the assembly of the rigid body modes of the individual elements taking into account the multiplicity of those degrees of freedom that lie in multiple neighboring elements."

does that really make sense?

• I'm not sure you can predict subdomains RBMs, but you can surely improve your conditioning by deflating on the linear space of RBMs for each subdomain. The nice thing is that the RBMs for elasticity are analytical so their computation from the elements is straightforward.
– Tom
Jan 5, 2015 at 14:56
• Tom, in order to deflate the CG, I need a vector, which shows the charactheristics of element nullspace vector and also its size should be compatible with solution size. I could not get it how to form it. Jan 5, 2015 at 16:16
• On each subdomain you have 6 vectors, so basically you have 6*nb subdmains deflation vectors. Or you can choose to assemble each singular vector and just keep 6 singular vector for all your structure. I used such deflation technique for Schartz additive preconditioner with many subdomains and it works quite well for elasticity.
– Tom
Jan 5, 2015 at 17:51