4301EllipticHigherD08

4301EllipticHigherD08 - Applied Mathematics 4301: Numerical...

Info iconThis preview shows pages 1–13. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Applied Mathematics 4301: Numerical Methods for PDEs Elliptic PDEs Lecture #5 (Part B) 1 October 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 Numerical procedure for PDEs Choose well-posed 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, self-adjoint 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 first-order convective terms, which spoil our self- adjointness and (possibly) our diagonal dominance [subsequent lecture] Finite elements a 30,000 ft view Residual K-dim 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 infinite-dimensional function spaces Greens theorem replaces 1D integration by parts Galerkin technique in finite-dimensional 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 ( x-y ) recovers strong Finite-dimensional 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 ], [ )...
View Full Document

This note was uploaded on 08/30/2011 for the course APMA 4301 taught by Professor Keyes during the Fall '08 term at Columbia.

Page1 / 79

4301EllipticHigherD08 - Applied Mathematics 4301: Numerical...

This preview shows document pages 1 - 13. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online