4301IntroToFE1

4301IntroToFE1 - complexity, FD is often used in time....

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Applied Mathematics 4301: Numerical Methods for PDEs Finite Elements and the Heat Equation Wed aft. 4:10-6:40 S. W. Mudd Bldg. 1024 Prof. David Keyes, instructor S. W. Mudd Bldg. 215 [email protected] Yan Yan, teaching assistant [email protected] Lecture #3 (Part C) 17 September 2008
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The versatile Laplacian • Aerospace engineer: flow potential • Mechanical engineer: temperature • Civil engineer: stress potential • Chemical engineer: concentration • Electrical engineer: electrical potential • Computational scientist: Goose the lays the golden eggs f u u u yy xx = + 2 To a …, u is a … (in 2D)
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The Laplacian is the spatial part of the parabolic equation u u u yy xx t + = Once we have a semi-discretization in space, we can use our catalog of time-discretization schemes to create complete discretizations. The space and time discretizations need not be of the same type. Even when FE is used in space to handle
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Unformatted text preview: complexity, FD is often used in time. Finite elements are used, in time, too, but this is much less standard. (in 2D) Meet finite elements (recap) • A few key contrasts between finite elements and finite differences are: – A finite element solution is defined everywhere, not just at certain points • This leads to a much richer theory – A finite element solution can be “weaker” than the continuous solution in the number of derivatives that it possesses • This leads to simplified algebraic and integration formulae – Finite elements can accommodate more general geometry • This makes them easier to generalize beyond simple test cases • For the 1D Laplacian, however, they look much the same! Finite elements Finite elements, cont. Finite elements, cont. x i x i+1 x i-1 Φ i (x) Φ i+1 (x) Finite elements, cont. Matrix formulation...
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This note was uploaded on 08/30/2011 for the course APMA 4301 taught by Professor Keyes during the Fall '08 term at Columbia.

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4301IntroToFE1 - complexity, FD is often used in time....

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