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The analysis in Section as applied to the model problem provides an attra ctive perspective to the solution of certain partial differential equations: the solution is identified w ith a “point”, which minimizes an appropriately constructed functional over an admis- sible function space.

**WEAK FORMULATI ON OF THE WEIGHTED RESIDUAL STATEMENT.**

The analysis in Section as applied to the model problem provides an attra ctive perspective to the solution of certain partial differential equations: the solution is identified w ith a “point”, which minimizes an appropriately constructed functional over an admis- sible function space. Weak (variational) forms can be made fully equivalent to respective strong forms , as evidenced in the discussion of the weighted resid ual methods, under certain smoothness assumptions. However, the equivalence between weak (variati onal) forms and variational principles is not gu aranteed: indeed, there exists no general method of construct-

ing functionals I [u], whos e extremization recovers a desired weak (variational) form. In this

sense, only certain partial d ifferential equations are amenable to analysis and solution by variational methods.

Vainberg’s theorem provides the necessary and sufficient condition for the equivalence of a weak (variational) form to a functi onal extremization problem. If such equivalenc e holds, the functional is referred to as a potential.

Theorem (Vainberg)

Consider a weak (variational) form

G(u, δu) := B(u, δu) + (f, δu) + (q¯ , δu)Γq = 0 ,

where u ∈ U , δu ∈ U0 , and f and q¯ are independent of u. Assume th at G pos- sesses a Gˆateaux derivative in a neighb orhood N of u, and the Gˆateaux differen- tial Dδu1 B(u, δu2) is

continuous in u at every point of N .

Then, the necessary and sufficient condition for the above weak form to b e derivable from a

potential in N is that

Dδu1 G(u, δu2) = Dδu2 G(u, δu1) ,

Namely that Dδu1 G(u, δu2) be symmetric for all δu1, δu2 = U0 and all u = N .

Preliminary to proving the above theorem, introduce the following two lemmas:

Lemma 1 Show that Dv I[u] = lim

In the above derivation, no te that operations and |ω=0 are not interchan geable (as they

both refer to the same variable ω), while lim∆ω→0 and |ω=0 are interchangeable, conditional upon sufficient smoothness of I [u].

Lemma 2 (Lagrange’s formula)

Let I [u] be a functional with Gateaux derivatives everywhere, and u, u + δu be any points of U. Then,

I [u + δu] − I [u ] = Dδu I [u + ǫ δu] 0 < ǫ < 1.

To prove Lemma 2, fix u and u + δu in U, and define function f on R as

f(ω) := I[u + ω δu] .

It follows that

Where Lemma 1 was inv oked. Then, u s i n g the standard mean-value theorem of calculus,

Tags : Mechanical - Finite Element Analysis - Finite Element Formulation of Boundary Value Problems

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