Introduction to the Finite Element Method - Lecture 02
Introduction to the Finite Element Method - Lecture 02
Introduction to the Finite Element Method - Lecture 02
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Function Spaces<br />
<strong>Lecture</strong> <strong>02</strong><br />
pk285@<br />
Boundary<br />
Value<br />
Problems<br />
<strong>Finite</strong> <strong>Element</strong><br />
<strong>Method</strong><br />
Function Spaces<br />
Weak Forms<br />
Since we are going <strong>to</strong> be approximating solutions of PDEs<br />
i.e. functions, it makes sense <strong>to</strong> recap some of <strong>the</strong> basic<br />
function spaces and <strong>the</strong>ir properties. If Ω is an open subset<br />
of R (or R n ) in general, <strong>the</strong>n:<br />
C(Ω) contains all functions defined on Ω which are<br />
continuous.<br />
C k (Ω) contains all functions defined on Ω which have<br />
continuous derivatives up <strong>to</strong> order k.<br />
Cb k(Ω) same as Ck (Ω) plus <strong>the</strong> function is bounded<br />
L 2 (Ω) contains all functions defined on Ω which are square<br />
integrable i.e.: ∫<br />
u 2 (x) dx < +∞ (6)<br />
Ω<br />
H 1 (Ω) contains all functions in L 2 whose derivatives are also<br />
square integrable i.e.:<br />
∫<br />
|du/dx| 2 (x) dx < +∞ (7)<br />
Ω
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