Domain embedding preconditioners for mixed systems (Q2716337)

The authors deal with block diagonal preconditioners for mixed systems derived from Dirichlet problems for second order elliptic equations. The main purpose is to discuss the question how an embedding of the original computational domain into a simpler extended domain can be utilized in this case. The authors show that a family of uniform preconditioners for the corresponding problem on the extended or fictitious domain leads directly to uniform preconditioners for the original problem. This is in contrast to the situation for the standard finite element method, where the domain embedding approach for the Dirichlet problem is less obvious. NEWLINENEWLINENEWLINEIn this approach the authors utilize an embedding of the original domain into an extended domain to construct a proper preconditioner. In practical applications the geometry of the extended domain might be simpler, or more regular, than the geometry of the original domain. NEWLINENEWLINENEWLINEThe introduction and the subsequent section of the paper are devoted to an introduction into the mixed finite element method for elliptic boundary value problems, while preconditioning of saddle point problems is discussed. The authors described the tight connection between the preconditioners for the indefinite mixed systems and the preconditioners for the positive definite operator derived from the inner product in the divergence space \({\mathbf H}(div)\) of vector valued functions. NEWLINENEWLINENEWLINEDomain embedding preconditioners for positive definite operator are discussed and the key tool in the analysis is the construction of an extension operator which is bounded in divergence space. In the following the authors show how the \({\mathbf H}(\text{div})\) preconditioner proposed here can be combined with the auxiliary space technique to construct preconditioners for systems obtained from the non-conforming Crouzeix-Raviart method by viewing this method as a non-conforming mixed method. Some numerical experiments are presented in the last part of the paper.