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For the functional  IL(v,λ)≡Iv(v)+∫ΩcλG(v)dxdy, what is the necessary condition for IL to have a stationary value?

(a) δvxIL+δvyIL+δλIL=0

(b) δvxIL-δvyIL-δλIL=0

(c) δvxIL-δvyIL+δλIL=0

(d) δvxIL+δvyIL-δλIL=0

I have been asked this question in final exam.

This intriguing question comes from Penalty in section Flows of Viscous Incompressible Fluids of Finite Element Method

1 Answer

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by (185k points)
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Best answer
Right choice is (a) δvxIL+δvyIL+δλIL=0

For explanation: In the Lagrange multiplier method the constrained problem is reformulated as one of finding the stationary points of the unconstrained functional  IL(v,λ)≡Iv(v)+∫ΩcλG(v)dxdy where λ(x,y) is the Lagrange multiplier. The necessary condition for IL to have stationary value is δvxIL+δvyIL+δλIL=0.

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