Some Domain Decomposition Methods for Discontinuous Coecients
Marcus V. Sarkis · 2011
Theory of Schwarz Methods Additive Schwarz Tas = ( ∑N i=0 R T i A −1 i Ri )A = (B)A Lower bound: C−2 0 a(u, u) ≤ a(Tasu, u) I Lions’ Lemma: Find a C0 > 0 such that, for any u ∈ V , there exists a decomposition u = ∑N i=0 R T i ui such that N ∑ i=0 ai (ui , ui ) ≤ C 2 0 a(u, u) Upper bound: a(Tasu, u) ≤ ωρ(E)a(u, u) I Inexact solvers: Find an ω where for any i = 0 : N and ui ∈ Vi a(R i ui ,R T i ui ) ≤ ωai (ui , ui ) I Strengthened Cauchy-Schwarz: Find a upper bound for the spectral radius of E = { ij}i,j=0:N a(R i ui ,R T j uj) ≤ ija(R i ui ,R i ui )a(R j uj ,R j uj) Marcus Sarkis (WPI) Domain Decomposition Methods RICAM-2011 24 / 38 Decomposition Partition of Unity φi (x) ∈ Vh(Ω) for the overlapping subdomains Ωi supp(φi ) ⊂ Ω δ i 0 ≤ φi (x) ≤ 1, x ∈ Ω δ i N ∑ i=1 φi (x) = 1, x ∈ Ω ‖∇φi‖∞ ≤ C/δ, 1 ≤ i ≤ N Decomposition of u = u0 + ∑N i=1 ui I u0 = I B h u. Let w = u − u0 I ui = Ih(φiw) Marcus Sarkis (WPI) Domain Decomposition Methods RICAM-2011 25 / 38 Lower bound estimation We need to estimate