Domain Decomposition Methods Sample Clauses

Domain Decomposition Methods. Another important class of methods we will use for the linear solver are domain de- composition methods. The concept will be explained following [90]. The first method based on a decomposition of the domain was introduced by Xxxxxxx in 1870. His method, referred to as the alternating Xxxxxxx method, can be used to solve elliptic boundary value problems on domains that are the union of two subdo- mains by alternatingly solving the elliptic boundary value problem restricted to the individual subdomains. Consider the domain Ω = Ω1 ∪ Ω2 as shown in Figure 6.2 on which we wish to solve the linear elliptic PDE  Lu = f in Ω, (6.4)  u = g on ∂Ω. We consider here Dirichlet boundary conditions, but other types of boundary Figure 6.2: Original figure by Xxxxxxx [90]. conditions are possible as well. The artificial boundaries are denoted by Γ1 and Γ2, i as shown in Figure 6.2. Let u(k) denote the approximate solution of (6.4) on the subdomain Ωi, i = 1, 2. We begin the alternating Xxxxxxx iterations by selecting an initial guess u(0) for the solution in subdomain Ω2. We then solve for k = 1, 2, ... the boundary value problem   Lu(k) u = f in Ω2