The Skeleton Reduction for Finite Element Substructuring Methods

We introduce an abstract concept for decomposing spaces with respect to a substructuring of a bounded domain. In this setting we define weakly conforming finite element approximations of quadratic minimization problems. Within a saddle point approach the reduction to symmetric positive Schur complement systems on the skeleton is analyzed. Applications include weakly conforming variants of least squares and minimal residuals. We consider general weakly conforming substructuring methods and its hybridization for the approximation of linear differential equations on Lipschitz domains Ω ⊂ R. The discretization is based on a decomposition Ωh = ⋃ K∈KK into convex open subdomains K ⊂ Ω with weak continuity constraints on the skeleton Γ = ⋃ ∂K = Ω \ Ωh. Here we present a general concept for the analysis of such discretizations based on corresponding saddle point formulations, and following [7] we consider the reduction to degrees of freedom to the skeleton. For comparison, we also summarize the DPG method [4] in this setting using formal trace mappings arising from integration by parts and quotient spaces replacing trace spaces. 1 Substructuring, trace spaces, and minimization Let L be a linear first-order differential operator with Lv ∈ L2(Ω,R ) for v ∈ C0(Ω,R ), and let L be its adjoint operator with (Lv,w)Ω = (v, L w)Ω , v ∈ C0(Ω,R ) , w ∈ C0(Ω,R ) . We define for L in Ω (and analogously for L and for open subsets of Ω) H(L,Ω) = { v ∈ L2(Ω,R ) : f ∈ L2(Ω,R ) exists with (f, w)Ω = (v, L w)Ω for all w ∈ C0(Ω,R ) } . Then, L extends to this space, and H(L,Ω) is a Hilbert spaces with respect to the graph norm ‖v‖L,Ω = √ ‖v‖Ω + ‖Lv‖Ω . For open subsets K ⊂ Ω we define the bilinear map γK(v, w) = (Lv,w)K − (v, Lw)K , v ∈ H(L,K) , w ∈ H(L,K)