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The main difference between the Rayleigh-Ritz method (RRM) and the finite element method lies in the definition of the basis functions. For FEM, these are element-related functions, whereas for RRM these are valid for the whole domain and have to fit the boundary conditions. The Rayleigh-Ritz method for homogeneous boundary conditions leads to the same discretized equations as the Galerkin method of weighted residuals.
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