4301IntroToFE2

4301IntroToFE2 - Lecture #4 (Part B) 24 September 2008...

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Applied Mathematics 4301: Numerical Methods for PDEs Finite Elements for the Steady and Transient Heat Equation Lecture #4 (Part B) 24 September 2008 Wed aft. 4:10-6:40 S. W. Mudd Bldg. 1024 Prof. David Keyes, instructor S. W. Mudd Bldg. 215 apam4301@gmail.com Yan Yan, teaching assistant yy2150@columbia.edu
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Let’s meet FE again • A few key contrasts between finite elements and finite differences: – 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 algebra and integration formulae – Finite elements can accommodate more general geometry • This makes them easier to generalize beyond simple test cases
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Plan of development • We start with the spatial (steady) term of the diffusion operator (Ch. 5 & 6, Gockenbach), for ODE BVPs, a 1D elliptic prototype – homogeneous Dirichlet BCs, constant coefficients – variable coefficients – inhomogeneous Dirichlet BCs – homogeneous Neumann BCs – inhomogeneous Neumann BCs • We then put back the temporal (transient) term (Ch. 6, Gockenbach), 1D parabolic – homogeneous Dirichlet BCs, constant coefficients – homogeneous Neumann BCs
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Finite Element procedure • Rewrite “strong” pointwise form of PDE in “weak” variational form – multiply PDE by an arbitrary test function (satisfying certain BCs) and integrate over the domain – instead of the PDE being valid at every one of an infinite number of points, the variational form must be valid for every one of an infinite number of test functions • Reduce the space of trial functions for solving the PDE and the space of test functions to something finite, storable, and computable • Choose a particular basis in which to expand the trial and test functions and derive algebraic equations for the coefficients of the expansion – typically, piecewise polynomials with local support are chosen – lesson from week #1: secret to a good life is picking the right basis
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• Strong form • Weak form δ ( x-y ) recovers strong • Finite-dimensional weak form • Algebraic system Ω = x S x u x f x u , ) ( ) ( ) ( L T x v S x u x x f x v x x u x v = Ω Ω ) ( , ) ( d ) ( ) ( d ) ( ) ( L } , 1 , { span , ) ( } , 1 , span{ , ) ( d ) ( ) ( d ) ( ) ( N j T T x v N i S S x u x x f x v x x u x v j h h i h h = = = = = Ω Ω ψ ϕ L ) , ( d ], [ ) , ( d ], [ , ) ( ) ( , , j j j j j i i j j i j i i i i f x f f f a x K K x u x u = = = Ω Ω f K f Ku L
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4301IntroToFE2 - Lecture #4 (Part B) 24 September 2008...

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