Read e-book online An introduction to G-convergence PDF

By Gianni Dal Maso

ISBN-10: 081763679X

ISBN-13: 9780817636791

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Endfor Compute: r(x) ≡ ∗ v1 (x) − v2 (x), for x ∈ Ω12 ∗ 0, for x ∈ Ω12 5. For i = 1, 2 in parallel solve the adjoint problems: ⎧ ∗ ⎪ ⎨ −∇ · (a ∇wi ) − ∇ · (b wi ) + c wi = r(x), in Ωi wi = 0, on B[i] ⎪ ⎩ ni · (a∇wi + b wi ) = 0, on B (i) . 6. 7. Endfor Update: (k+1) (x) = g1 (x) − τ w1 (x), for x ∈ B (1) (k+1) (x) = g2 (x) + τ w2 (x), for x ∈ B (2) . g1 g2 8. 33) using its saddle point formulation. However, the resulting algorithm may require more computational resources. For instance, suppose that: J(v1 , v2 ) = 1 v1 − v2 2 2 ∗ ), L2 (Ω12 and that Neumann boundary conditions are imposed on B (i) .

However, care must be exercised in discretizing the transmission conditions so that the resulting global discretization is stable. 19) using finite element methods. ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ ❜ r r r r r❜ r r❜ r r❜ r❜ r r❜ ❜ ❜ r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r Th2 (Ω2 ) Th1 (Ω1 ) Fig.

Then u(x) = w1 (x) in Ω 1 and u(x) = w2 (x) in Ω 2 . Proof. 19) for the substitution µ = n2 · (a∇u) on B. 26. 30). 30). 29) associated with it [GI3]. 23) can be obtained by applying a saddle point iterative algorithm such as Uzawa’s method, see Chap. 10, to update the Lagrange multiplier function µ(·), as described below. 1 (Uzawa’s Method) Let µ(0) denote a starting guess with chosen step size τ > 0. 1. For k = 0, 1, · · · until convergence do: (k+1) (k+1) and w2 in parallel: 2. Determine w1 ⎧ (k+1) (k+1) ⎪ −∇ · a∇w1 + c w1 = f, in Ω1 ⎪ ⎪ ⎪ ⎪ (k+1) ⎪ ⎪ w1 = 0, on B[1] ⎪ ⎪ ⎪ (k+1) ⎪ (k) ⎪ = −µ , on B, n1 · a∇w1 ⎨ ⎪ (k+1) (k+1) ⎪ ⎪ + c w2 = f, −∇ · a∇w2 ⎪ ⎪ ⎪ ⎪ (k+1) ⎪ w2 = 0, ⎪ ⎪ ⎪ ⎪ (k+1) ⎩ = µ(k) , n2 · a∇w2 3.

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An introduction to G-convergence by Gianni Dal Maso


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