Masters Thesis - TU Delft

5.3 Schur ComplementLet’s consider a following problem:Lu = f in Ωu = g on ∂Ω (5.8)with the domain Ω partitioned onto s subdomains. After discretization of the problem, wecan label the nodes by subdomain in a specific way, so the linear system will have a followingstructure:⎡ ⎤⎡B 1 E 1B 2 E 2. .. ⎢.⎥⎢⎣ E s⎦⎣F 1 F 2 ... F s Cx 1x 2.x sy⎤⎡=⎥ ⎢⎦ ⎣where each x i represents the subvector of unknowns that are interior to subdomain Ω i , and yrepresents the vector of all interface unknowns. It is useful to write the system in a more simpleform, i.e.[ ] [ ] x fA = ,where A =y g[ B EF Cf 1f 2.f sg]⎤⎥⎦(5.9)(5.10)and where E represents the subbdomain to interface coupling seen from the subdomains, whileF represents the interface to subdomain coupling seen from the interface nodes. To illustratethis, let us consider a domain split into only two subdomains. Let’s assume that the subdomainsare of the same size and both are squared. Then an illustrative mesh and correspondingcoefficient matrix A will look likeFigure 5.2: An exemplary mesh for the problem described above.24