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...
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 Fall '10
 Morel
 Numerical Analysis, Partial differential equation, finite difference, discretization, approximate solution

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