ABSTRACT
This chapter discusses the more general high-order polynomial shape functions, which may be readily applied without any expectation of the solution. Since the solution space is already approximate and of limited dimension, strongly asserted conditions further reduce the freedom of the solution to fit the governing equation and are generally considered undesirable. The chapter demonstrates that the implementation on the variational formulation, extensions to the Galerkin method being straightforward. It describes pre-computed universal matrices, which allow to deal with higher-order finite elements using trivial computations that compare in difficulty only with those required for first-order finite elements. The Galerkin method is a powerful scheme suited to many a differential equation especially when there is no functional identified with that equation. The Galerkin method provides with the best of the various solutions within the approximation space. Thus, the approximate solution is limited by the trial function and gives the best-fit of the governing equation from the trial space.
