4301Intro20080924

4301Intro20080924 - Lecture #4 (Part A) 24 September 2008...

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Applied Mathematics 4301: Numerical Methods for PDEs Introduction, Lecture #4 Wed aft. 4:10-6:40 S. W. Mudd Bldg. 535 Prof. David Keyes, instructor S. W. Mudd Bldg. 215 [email protected] Yan Yan, teaching assistant [email protected] Lecture #4 (Part A) 24 September 2008
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Plan for Sep 24 meeting • Review • Questions <break> • Main lecture: “Finite elements for the steady and transient heat equation” (from chapters 5 & 6 of Gockenbach)
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Review of Sep 17 meeting • Model parabolic problem for transient diffusion in higher dimensions – linear, homogeneous, Dirichlet – with eigenfunction expansion solution • Explicit FD method – discretization & marching solution – truncation error – stability and convergence • Implicit and Weighted FD methods – discretization and dimensional splittings – stability of various splitting algorithms • Generalizations of the model problem – complex geometry – anisotropy
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Question on error propagation • Begin with definition of the scheme • Then the definition of the truncation error • Then the definition of the pointwise error • Subtract (2) from (1) 2 1 1 1 2 1 1 1 2 1 1 1 ) 2 ( ) , ( ) , ( )) , ( ) , ( 2 ) , ( ( ) , ( ) , ( ) , ( ) 2 ( 0 h e e e k e e t x T t x u U e h t x u t x u t x u k t x u t x u t x T h U U U k U U n j n j n j n j n j n j n j n j n j n j n j n j n j n j n j n j n j n j n j n j + + + + + + + = + = + =
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Convergence proof from error prop.
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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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4301Intro20080924 - Lecture #4 (Part A) 24 September 2008...

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