Applied Computational Fluid Dynamics Techniques: An Introduction Based on Finite Element Methods, 2nd Edition

5 APPROXIMATION OF OPERATORS

While approximation theory dealt with the problem

the numerical solution of differential or integral equations deals with the following problem:

Here L(u) denotes an operator, e.g. the Laplace operator L(u) = ²u. The aim is to minimize the error of the operator using known functions

As before, the method of weighted residuals represents the most general way in which this minimization may be accomplished

5.1. Taxonomy of methods

The choice of trial and test functions Nⁱ, Wⁱ defines the method. Since a large amount of work has been devoted to some of the more successful combinations of Nⁱ, Wⁱ, a classification is necessary.

5.1.1. FINITE DIFFERENCE METHODS

Finite difference methods (FDMs) are obtained by taking Nⁱ polynomial, Wⁱ = δ(x_i). This implies that for such methods is enforced at a finite number of locations in space. The choices of polynomials ...