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nodal variables is much more di cult. Suppose, for example, there are l boundary
conditions of the form Tc = (5.5.15) where T is an l N matrix and is an l-vector. In vector systems of partial di erential
equations, such boundary conditions arise when constraints are speci ed between di erent
components of the solution vector. In scalar problems, conditions having the form (5.5.15)
arise when a \global" boundary condition like
Z @ uds = is speci ed. They could also arise with periodic boundary conditions which might, for
example, specify u(0 y) = u(1 y) if u were periodic in x on a rectangle of unit length.
One could possibly solve (5.5.15) for l values of ci, i = 1 2 ::: N , in terms of the
others. Sometimes there is an obvious choice however, often there is no clear way to
choose the unknowns to eliminate. A poor choice can lead to ill-conditioning of the
algebraic system. An alternate way of treating problems with boundary conditions such
as (5.5.15) is to embed Problem (5.5.11) in a constrained minimization problem which
may be solved using Lagrange multipliers. Assuming K to be symmetric and positive
semi-de nite, (5.5.11) can be regarded as the minimum of I c] = cT Kc ; 2cT f :
Using Lagrange multipliers, we minimize the modi ed functional
I c ] = cT Kc ; 2cT f + 2 T (Tc ; ) 5.5. Element Matrices and Their Assembly
yields 33 ~
is an l-vector of Lagrange multipliers. Minimizing I with respect to c and KT
T0 c=f: T (5.5.16) The system (5.5.16) may or may not be simple to solve. If K is non-singular then the
algorithm described in Problem 2 at the end of this section is e ective. However, since
boundary conditions are prescribed by (5.5.15), K may not be invertible.
Nontrivial Neumann boundary conditions on @ N require the evaluation of an extra
line integral for those elements having edges on @ N . Suppose, for example, that the
variational principle (5.5.1) is replaced by: determine u 2 HE satisfying A(v u) = (v f )+ < v >
where < v >= Z
8v 2 H0 v (x y)ds (5.5.17a)
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This document was uploaded on 03/16/2014 for the course CSCI 6860 at Rensselaer Polytechnic Institute.
- Spring '14