# In the penalty formulation of the fluid flow model, if the velocity field (vx, vy ) satisfies the continuity equation, then the weight functions (w1, w2) also satisfy the continuity equation.

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In the penalty formulation of the fluid flow model, if the velocity field (vx, vy ) satisfies the continuity equation, then the weight functions (w1, w2) also satisfy the continuity equation.

(a) True

(b) False

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

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The correct option is (b) False

To elaborate: In the penalty formulation, we begin with the unconstrained problem described by the weak forms of the mixed model, without the time—derivative terms. Now, suppose that the velocity field (vx, vy )  is such that the continuity equation is satisfied identically. Then the weight functions (w1, w2) being (virtual) variations of the velocity components, also satisfy the continuity equation.

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