The Finite Element Method: Theory, Implementation, and Applications by Mats G. Larson & Fredrik Bengzon

Author:Mats G. Larson & Fredrik Bengzon
Language: eng
Format: epub
Publisher: Springer Berlin Heidelberg, Berlin, Heidelberg

(7.71)

implying

(7.72)

for each v ∈ V. This is exactly our abstract variational problem.

Now, let u instead be the solution to the abstract variational problem (7.60). Expanding F(u + v), with v ∈ V arbitrary, we have

(7.73)

(7.74)

(7.75)

(7.76)

(7.77)

where we have used that a(u, v) = l(v) for all v ∈ V. From this we conclude that F(u + w) attains its minimum value for v = 0.

Hence, (7.60) and (7.66) are equivalent. □

In mechanics applications the minimization problem can be through of as a demand for minimal energy. Indeed, the functional F(⋅) represents the mechanical energy of the physical system under consideration, and the solution u a displacement from a state of rest into one of equilibrium of forces.

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