Specific Examples of the Resolution of the Poisson's equation with the WRM using global Shape Functions
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(Difference between revisions)
(→Least Squares Method <math>(W_i(x)=A(\varphi) \quad and \quad \overline{W_i}(x)=B(\varphi))</math>) 

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:[[Poisson's_Equation#Weighted_Residual_Method_.28WRM.29  ''Back to the Weighted Residual Method applied to the Poisson's equation'']]  :[[Poisson's_Equation#Weighted_Residual_Method_.28WRM.29  ''Back to the Weighted Residual Method applied to the Poisson's equation'']]  
−  [[Category:  +  [[Category:Poisson's Equation]] 
Latest revision as of 09:08, 4 November 2009
 The matrix form of the residual formulation can be written as: with:
 For simplicity reasons, we select as basis functions the Fourier Series , that automatically accomplish the boundary conditions and . If the material is constant, we can write:
sources medium boundary conditions
shape functions  constant source
 homogeneous
 Dirichlet
 Fourier Series
 By using different weighting functions, a new range of methods can be classified:
Contents 
Collocation Method
General expression  k and Q constant, using Fourier series 



 Number of points = 1
 Number of points = 2
 Number of points = n
 To modify the parameters, edit this Matlab code
Subdomain Method
General expression  k and Q constant, using Fourier series 



 Number of points = 1
 Number of points = 2
 Number of points = n
 To modify the parameters, edit this Matlab code
Galerkin Method
General expression  k and Q constant, using Fourier series 



 Number of points = 1
 Number of points = 2
 Number of points = n
 To modify the parameters, edit this Matlab code
Least Squares Method
General expression  k and Q constant, using Fourier series 



 Note that, in this case and without generality, due to the selected basis functions, the obtained solution is the same that the obtained one using the Galerkin method scaled with a factor .
 To modify the parameters, edit this Matlab code