This preview shows page 1. Sign up to view the full content.
Unformatted text preview: only if is a solution of the equations. This leads us to the central theme of finite element methods. Let us assume that we want to solve and that we have a basis function representation for our approximate solution: ̃ ∑ ( ⃗) The residual associated with this representation for ̃ is
( ̃ ) Since ̃ is an approximate solution, we cannot expect to make the residual exactly zero. Rather, the
central theme of finite element methods is to make the residual “small” in some sense. For the case of a
linear operator (equation), we can show an exact relationship between the residual and the additive
error associated with the approximate solution. In particular, the exact solution satisfies While the approximate solution satisfies
̃ ( ̃) Subtracting the bottom equation from the top equation, and assuming linearity, we get
The additive error in the approximate solution is ̃) ( ̃) ̃
If we solve the equation for the error, we get
Thus the error in the approximate solution satisfies th...
View Full Document
This note was uploaded on 10/06/2013 for the course NEUN 430 taught by Professor Morel during the Fall '10 term at Texas A&M.
- Fall '10