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By Henning F. Harmuth

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As a consequence of the transformation theorem for integrals, we find the correct transformation DTΦ v = | det DΦ|(v ◦ Φ) . Here, DΦ : Ω → Rd,d stands for the Jacobi matrix of Φ. 30 (DT) 2 Elliptic Boundary Value Problems Another important class of functions comprises gradient type vectorfields u : Ω → Rd . For them the appropriate transformation reads GTΦ u := DΦT (u ◦ Φ) . (GT) The next result underscores that (GT) is the correct transformation of gradients. 6. t. the -variables. Proof. A straightforward application of the chain rules confirms ∂ ∂ ξk u(Φ(ξ)) = grad u(Φ(ξ) , ∂Φ , ∂ ξk which amounts to the assertion of the lemma written componentwise.

The classification clearly hinges on the variational formulation. Consider the boundary value problem (E2D) with ΓN = Γ. This will be related to the global minimization of the functional J(v) := 1 2 Ω c−1 | div v|2 + A−1 |v|2 dξ − c−1 f div v dξ Ω over the space Xh := {v ∈ (C 1 (Ω))3 ∩ (C 0 (Ω))3 , v, n = h} . This gives rise to the variational problem: find j ∈ X−h such that c−1 div j div v + A−1 j, v dξ = Ω c−1 f div v dξ Ω ∀v ∈ X0 . (FVD) Obviously, the Neumann boundary conditions turn out to be essential boundary conditions in this case.

6: Distances tj . Notation: Writing γ(Ω) we hint that the “constant” γ may only depend on the domain Ω. Proof. The proof will be presented for a convex polygon Ω only: denote by ξ˜ the center of the largest d−dimensional ball inscribed into Ω and by ρΩ its radius. Without loss of generality, we suppose that ξ˜ is the origin of the coordinate system. We start from the following relation ∂Ω v 2 ξ · n dS = div(v 2 ξ) dξ, Ω v ∈ H 1 (Ω). 27) Let nj be the outer unit normal to Ω on the edge Γj , j ∈ S.

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