[PDF] finite element method basis functions

What is the finite element method?

The Finite Element Method provides a general and systematic technique for constructing basis functions for Galerkin's approximation of boundary value problems. The idea of finite elements is to choose piecewise over subregions of the domain called finite elements. Such functions can be very simple, for example, polynomials of low degree.

What is a finite element basis function?

The finite element basis functions ?i are now defined as follows. and zero at all other nodes in the element. On all other elements, ?i = 0 . . On all other elements, ?i = 0 . A visual impression of three such basis functions are given in Figure Illustration of the piecewise quadratic basis functions associated with nodes in element 1.

Why do we use finite elements in FEM?

When more nodes are used the approximation improves and in the limit of N? ? the sine wave would be exactly represented. In FEM we will never proceed all the way to the limit, so the interval size hwill always have ?nite size h. This is why the term ?nite elements is used. 0 0.5 1 1.5 2 2.5 3 3.5 4 ?1 0 1 h 1 h 2 h 3 h 4 ? 1 ? 2 ? 3 4 0.7071? 1 ? 2

What are the two types of variables in the finite element method?

Therefore, the resulting initial/boundary-value problems include two types of independent variables, namely, spatial variables (e.g., xi, i= 1,2,3) and a temporal variable (t). In the context of the ?nite element method, there are two general approaches in dealing with the two types of variables.