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Unformatted text preview: Applied Mathematics 4301: Numerical Methods for PDEs Elliptic PDEs Lecture #5 (Part B) 1 October 2008 Wed aft. 4:106:40 S. W. Mudd Bldg. 1024 Prof. David Keyes, instructor S. W. Mudd Bldg. 215 apam4301@gmail.com Yan Yan, teaching assistant yy2150@columbia.edu Numerical procedure for PDEs Choose wellposed continuous model and formulation existence, uniqueness, conditioning Choose discretization grid, method, basis, order, etc. Choose solution procedure Iteratively : solve estimate errors interpret results if necessary, improve model, mesh, etc. General Elliptic BVPs We have looked at the constant coefficient, selfadjoint 1D ODE BVP rather thoroughly in Fourier spectral, FD, and FE, for various BC types We should move to higher dimensions for FE and FD [tonight] We should return to 1D and add firstorder convective terms, which spoil our self adjointness and (possibly) our diagonal dominance [subsequent lecture] Finite elements a 30,000 ft view Residual Kdim expansion Weighted residual Collocation Moment Galerkin Least squares Energy = = = = = = = dx x u x u x u x f dx x r x x w x x w x x x w K k dx x r x w x a x u x u f(x) x r k k a a k k k k k k k K k k k )] ( ) ( 2 1 ) ( ) ( [ min  ) (  min ) ( ) ( ) ( ) ( ) ( , , 2 , 1 , ) ( ) ( ) ( ) ( ) ( ) ( } { 2 } { ) 1 ( 1 L L L FE in higher spatial dimensions Variational form in infinitedimensional function spaces Greens theorem replaces 1D integration by parts Galerkin technique in finitedimensional function spaces basis functions are more complex to describe linear algebra problem has more complex sparsity structure Otherwise, identical machinery! Gockenbach, Section 8.4 Elliptic operators in higher dimensions Coordinate invariant form Cartesian, const. coeff. Heterogeneous material coeff Cartesian, var. coeff. Anisotropic material coeff Cartesian, const. coeff. ) , ( ) 2 ( ) ( ) , ( ) ) ( ) ) , ( ( ) , ( ) ( 22 12 11 y x f u u u f u y x f u u u u f u y x y x f u u f u u yy xy xx y y x x yy xx yy xx = + + = = + = = + = = t Variational development Variational development Greens identity IBP in higher D for the divergence of a product of scalar with a vector (in turn, a gradient): Greens identity Finite Element procedure FE procedure (review, for notation) Strong form Weak form ( xy ) recovers strong Finitedimensional weak form Algebraic system = 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 = ) ( , ) ( 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 = = = = = L ) , ( d ], [ )...
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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.
 Fall '08
 Keyes

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