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Finite Element Method
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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?

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+1 vote

asked Jan 30, 2022 in Finite Element Method by Ava (110k points)

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

finite element method
flows of viscous incompressible fluids
penalty

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1 Answer

+2 votes

answered Jan 30, 2022 by James (185k points)
selected Feb 3, 2022 by Ava

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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