14.4 Polynomial Interpolation

Suppose that at a finite number of points x _i in the domain ?, the function values u _i = u( x _i) are known, then a polynomial approximation, denoted by u _h, of u( x) can be constructed. The polynomial approximation u _h of degree n ? 1 can be constructed by

if u is known at n points. The coefficients a _i can be identified uniquely and expressed in terms of u _i, by solving the set of equations:

An example is given in Fig. 14.2, where, in the domain x _{i ?1} ? x ? x _{i + 2}, a thirdorder polynomial (dashed curve) is used to approximate a given function (solid curve) based on the function values u _{i ? 1}, u _i, u _{i + 1} and u _{i + 2}.

Figure 14.2: Solid line: u(x), dashed line: polynomial approximation of u(x).

Clearly, the coefficients a _i are linearly dependent on the values u _i, therefore the polynomial may be rewritten in terms of u _i by

where the functions N _i( x) are polynomial expressions of order n ? 1 in terms of the coordinate x. These functions N _i( x) are called shape functions because they define the shape of the interpolation of u _h, for instance linear, quadratic etc.