I know that a lot of topics within FEM have already been dealt within in here. However, I myself still need the big picture of FEM, and I were not able to retrieve it from the questions already being answered here, nor the common text books, which tends to be very technical from the first page, making it very inaccessible to first time learners.

As I understand it, the WRM (where the Galerkin method is one of several WRM) and the orthogonal collocation method is two different methods for approximating PDE's by piecewise functions, where the difference between the approximated and the 'true' solutions is minimized at certain spatial coordinates called collocation points. But what are the difference between these two? (ie. WRM and orthogonal collocation).

I have read that the collocation points are chosen as the roots of the "appropriate Jacobi polynomials". Why is it so?

Best regards

Edit: Edited according to Yuriy S's comment.

  • $\begingroup$ I think the question is a little too broad. You are better off with leaving the last part only (starting with "more detailed questions"). In that case the title should be remade too $\endgroup$ – Yuriy S Jan 17 '18 at 13:24

The Method of Weighted Residuals (MWR) or WRM is a family of methods for solving differential equations. Galerkin and collocation are two members of the family. Others are the method of moments, least squares, subdomain. MWR is based on the principal that weighted averages of the residual should vanish. The type of weight determines the individual method. Galerkin method weights by the trial functions. With collocation the weights are dirac delta functions. I have a paper coming out called "Orthogonal Collocation Revisited". It has a short history of these methods. It can be downloaded for free for a short time at - https://authors.elsevier.com/a/1YHLy_12dr4lJw . The paper also explains why the collocation points are chosen to be roots of Jacobi polynomials.


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