Stability and Oscillation Characteristics of Finite-Element, Finite-Difference, and Weighted-Residuals Methods for Transient Two-dimensional Heat Conduction in Solids

The finite-element difference expression was derived by use of the variational principle and finite-element synthesis. Several ordinary finite-difference formulae for the La-placian term were considered. A particular finite-difference formula for the Laplacian term was chosen to bring the difference expressions of finite-element, finite-difference, and weighted-residuals (Galerkin) methods into the same format. The stability criteria were established for all three techniques by use of the general stability, von Neumann, and Dusinberre concepts. The oscillation characteristics were derived for all three techniques. The finite-element method is more conservative than the finite-difference method, but not so conservative as the Galerkin method in both stability and oscillation characteristics.