In elastic problem, we often solve K * u = f, where K is the stiffness matrix, f the external force vector and u the displacement vector.
I'm trying decompose the mesh to domains, using Lagrange multipliers to enforce the displacement continuity on the interface between domains. Then for every domain, we have:
K_s * u_s = f_s + B_s * lambda
lambda is the vector of the Lagrange multipliers, and B is a signed boolean matrix to pick out the multipliers related to domain s.
Thus K * u = f turns to
|K1 B1| |u1| |f1| | K2 B2| |u2| |f2| | K3 B3| * |u3| = |f3| | K4 B4| |u4| |f4| |B1' B2' B3' B4' | |L | |0 |
here B1' mean the transpose of B1, L is the lambda vector.
When more than two domains intersect at a point, the multipliers at that point will be redundant:
1 | | 2 ------ ------- x ------ ------- 3 | | 4
Like in the above condition, four domains intersect at the point X, and it needs 6 multipliers to enforce continuity: 1-2, 1-3, 1-4, 2-3, 2-4, 3-4. Write in row vector form:
1 2 3 4 -------------- +1 -1 +1 -1 +1 -1 +1 -1 +1 -1 +1 -1
It's an example of rows of (B1' B2' B3' B4'). The problem is that these rows are linear dependent, only three of them can be linear independent, makes the whole matrix singular.
I know there's methods like FETI, which uses iterative methods to solve such problem. But I want to know whether there are some direct methods, for factorizing is much fast than PCG. I found a lot math papers on Lagrange multiplier based domain decomposed problem, but all of them discussing iterative method. Is there any work on direct method of this problem?