Error Inhibiting Methods for Finite Elements

Abstract

Finite Difference methods (FD) are one of the oldest and simplest methods used for solving differential equations. Theoretical results have been obtained during the last six decades regarding the accuracy, stability and convergence of the FD method for partial differential equations (PDE).The local truncation error is defined by applying the difference operator to the exact solution u. In the classical FD method, the orders of the global error and the truncation error are the same.Block Finite Difference methods (BFD) are difference methods in which the domain is divided into blocks, or cells, containing two or more grid points with a different scheme used for each grid point, unlike the standard FD method. In this approach, the interaction between the different truncation errors and the dynamics of the scheme may prevent the error from growing, hence error reduction is obtained. The phenomenon in which the order of the global error is smaller than the one of the truncation error is called error inhibition, see e.g. [8].The Finite Element method (FE) consists in finding an approximation of the solution in a certain form, usually a linear combination of a set of chosen trial functions, i.e.q j ϕ j . Since, in most cases, the exact solution will not lie in the space spanned by those functions, the coefficients of the