Galerkin method

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Artigo sobre o método de Galerkin

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Galerkin method

  1. 1. Computational Engineering Galerkin Method Yijian Zhan Ning Ma
  2. 2. Galerkin Method  Engineering problems: differential equations with boundary conditions. Generally denoted as: D(U)=0; B(U)=0  Our task: to find the function U which satisfies the given differential equations and boundary conditions.  Reality: difficult, even impossible to solve the problem analytically
  3. 3. Galerkin Method  In practical cases we often apply approximation.  One of the approximation methods: Galerkin Method, invented by Russian mathematician Boris Grigoryevich Galerkin.
  4. 4. Galerkin Method Related knowledge  Inner product of functions  Basis of a vector space of functions
  5. 5. Galerkin Method Inner product  Inner product of two functions in a certain domain: shows the inner product of f(x) and g(x) on the interval [ a, b ]. *One important property: orthogonality If , f and g are orthogonal to each other; **If for arbitrary w(x), =0, f(x) 0 , ( ) ( ) b a f g f x g x dx   , 0f g  ,w f  
  6. 6. Galerkin Method Basis of a space  V: a function space  Basis of V: a set of linear independent functions Any function could be uniquely written as the linear combination of the basis: 0{ ( )}i iS x   ( )f x V 0 ( ) ( )j j j f x c x    
  7. 7. Galerkin Method Weighted residual methods  A weighted residual method uses a finite number of functions .  The differential equation of the problem is D(U)=0 on the boundary B(U), for example: on B[U]=[a,b]. where “L” is a differential operator and “f” is a given function. We have to solve the D.E. to obtain U. 0{ ( )}n i ix  ( ) ( ( )) ( ) 0D U L U x f x  
  8. 8. Galerkin method Weighted residual  Step 1. Introduce a “trial solution” of U: to replace U(x) : finite number of basis functions : unknown coefficients * Residual is defined as: 0 1 ( ) ( ) ( ) n j j j U u x x c x       jc ( )j x ( ) [ ( )] [ ( )] ( )R x D u x L u x f x  
  9. 9. Galerkin Method Weighted residual  Step 2. Choose “arbitrary” “weight functions” w(x), let: With the concepts of “inner product” and “orthogonality”, we have: The inner product of the weight function and the residual is zero, which means that the trial function partially satisfies the problem. So, our goal: to construct such u(x) , ( ) , ( ) ( ){ [ ( )]} 0 b a w R x w D u w x D u x dx   
  10. 10. Galerkin Method Weighted residual  Step 3. Galerkin weighted residual method: choose weight function w from the basis functions , then These are a set of n-order linear equations. Solve it, obtain all of the coefficients . j 0 1 , [ ( )] ( ){ [ ( ) ( )]} 0 nb j j j ja j w R D u dx x D x c x dx           jc
  11. 11. Galerkin Method Weighted residual  Step 4. The “trial solution” is the approximation solution we want. 0 1 ( ) ( ) ( ) n j j j u x x c x     
  12. 12. Galerkin Method Example  Solve the differential equation: with the boundary condition: ( ( )) ''( ) ( ) 2 (1 ) 0D y x y x y x x x     (0) 0, (1) 0y y 
  13. 13. Galerkin Method Example  Step 1. Choose trial function: We make n=3, and 0 1 ( ) ( ) ( ) n i i i y x x c x      0 1 2 2 2 3 3 3 0, ( 1), ( 1) ( 1) x x x x x x           
  14. 14. Galerkin Method Example  Step 2. The “weight functions” are the same as the basis functions Step 3. Substitute the trial function y(x) into i 0 1 , [ ( )] ( ){ [ ( ) ( )]} 0 nb j j j ja j w R D u dx x D x c x dx          
  15. 15. Galerkin Method Example  Step 4. i=1,2,3; we have three equations with three unknown coefficients1 2 3, ,c c c 31 2 31 2 31 2 43 51 0 15 10 84 315 615 111 0 70 84 630 13860 734 611 0 315 315 13860 60060 cc c cc c cc c              
  16. 16. Galerkin Method Example  Step 5. Solve this linear equation set, get: Obtain the approximation solution 1 2 3 1370 0.18521 7397 50688 0.185203 273689 132 0.00626989 21053 c c c           3 1 ( ) ( )i i i y x c x   
  17. 17. Galerkin Method Example GalerkinGalerkin solutionsolution Analytic solutionAnalytic solution
  18. 18. References  1. O. C. Zienkiewicz, R. L. Taylor, Finite Element Method, Vol 1, The Basis, 2000  2. Galerkin method, Wikipedia: http://en.wikipedia.org/wiki/Galerkin_method#cite_note-BrennerScott-1

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