Computational Mechanics

, 39:91 | Cite as

Nodally exact Ritz discretizations of 1D diffusion–absorption and Helmholtz equations by variational FIC and modified equation methods

  • C. A. FelippaEmail author
  • E. Oñate
Original Paper


This article presents the first application of the Finite Calculus (FIC) in a Ritz-FEM variational framework. FIC provides a steplength parametrization of mesh dimensions, which is used to modify the shape functions. This approach is applied to the FEM discretization of the steady-state, one-dimensional, diffusion–absorption and Helmholtz equations. Parametrized linear shape functions are directly inserted into a FIC functional. The resulting Ritz-FIC equations are symmetric and carry a element-level free parameter coming from the function modification process. Both constant- and variable-coefficient cases are studied. It is shown that the parameter can be used to produce nodally exact solutions for the constant coefficient case. The optimal value is found by matching the finite-order modified differential equation (FOMoDE) of the Ritz-FIC equations with the original field equation. The inclusion of the Ritz-FIC models in the context of templates is examined. This inclusion shows that there is an infinite number of nodally exact models for the constant coefficient case. The ingredients of these methods (FIC, Ritz, MoDE and templates) can be extended to multiple dimensions


Finite calculus Variational principles Ritz method Functional modification Stabilization Finite element Diffusion Absorption Helmholtz Nodally exact solution Modified differential equation Templates 


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Copyright information

© Springer-Verlag 2005

Authors and Affiliations

  1. 1.Department of Aerospace Engineering Sciences and Center for Aerospace StructuresUniversity of ColoradoBoulderUSA
  2. 2.International Center for Numerical Methods in Engineering (CIMNE)Universidad Politécnica de CataluñaBarcelonaSpain

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