# In finite element modeling, which formulation introduces constraints on variables and satisfies them in an approximate sense?

In finite element modeling, which formulation introduces constraints on variables and satisfies them in an approximate sense?

(a) Velocity-pressure formulation

(b) Penalty formulation

(c) Mixed formulation

(d) Lagrange multiplier formulation

My question is taken from Penalty topic in section Flows of Viscous Incompressible Fluids of Finite Element Method

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The correct choice is (b) Penalty formulation

Easy explanation: There are two different finite element models of momentum and continuity equations. The first one is a natural and direct formulation of momentum and continuity equations. The second model, penalty formulation, is based on the interpretation that the continuity equation is a new relationship among the velocity components, and it eliminates the pressure term leading to a constraint on the velocity components. The constraint is satisfied in a least-squares (i.e., approximate) sense.

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