Thank you Wolfgang for the response. This cleared things up a bit. Den onsdag 24 augusti 2016 kl. 17:22:38 UTC+2 skrev Wolfgang Bangerth: > > > Anders, > > > I'm solving a 2 by 2 block system on the form > > [A -B' ; B A] * [y ; u] = [c ; d] > > where the matrix B is a stiffness matrix from a Laplace operator, > > −∆y=u, > > when I set Dirichlet BC on y (or u, or both) the corresponding rows in B > > are set to zero, even the diagonal element. > > Correct. The algorithm just looks at the overall matrix (with row and > column space defined by the DoFHandler). That you subdivide the matrix > somehow or other is not of concern to the algorithm. > > > > So, my question is if this is expected? Since this makes B singular > > which should not be the case for a stiffness matrix? The reason this is > > an issue is that the block B is used in a solver when preconditioning. > > Well, B is not the stiffness matrix. It is the matrix that results from > the operator > (nabla phi_i, nabla psi_j) > where phi_i and psi_j are the trial and test functions of the two > variables. It would be easier to see that it can't be *exactly* the > stiffness matrix if you imagined using different function spaces for y > and u (think, using Q2 elements for y, and Q1 elements for u). In your > case, it is also the matrix that is formed by considering *all* shape > functions, including those at the boundary, as test functions, but we > will have to eliminate these. > > > > My guess is that this keeps the diagonal of A and sets the rest of the > > corresponding row to zero, including B. Is there some other way to set > > the BC to avoid making B singular? > > No. Because then the linear system that results would not solve the > problem you want to solve. The matrix really needs to look like it > looks, with B singular. > > But what you could consider is not using B as the preconditioner, but > some "related" \tilde B. This matrix could, for example, have ones on > the diagonal. You wouldn't use it for forming the product with your > overall matrix in GMRES or whatever other method you use, but you'd use > it when preconditioning. > > Best > W. > > -- > ------------------------------------------------------------------------ > Wolfgang Bangerth email: [email protected] > <javascript:> > www: http://www.math.colostate.edu/~bangerth/ >
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