Piecewise Polynomial Approximation in 1D
In this chapter we introduce a type of functions called piecewise polynomials that can be used to approximate other more general functions, and which are easy to implement in computer software. For computing piecewise polynomial approximations we present
- PDF / 440,034 Bytes
- 22 Pages / 439.36 x 666.15 pts Page_size
- 72 Downloads / 272 Views
Piecewise Polynomial Approximation in 1D
Abstract In this chapter we introduce a type of functions called piecewise polynomials that can be used to approximate other more general functions, and which are easy to implement in computer software. For computing piecewise polynomial approximations we present two techniques, interpolation and L2 -projection. We also prove estimates for the error in these approximations.
1.1 Piecewise Polynomial Spaces 1.1.1 The Space of Linear Polynomials Let I D Œx0 ; x1 be an interval on the real axis and let P1 .I / denote the vector space of linear functions on I , defined by P1 .I / D fv W v.x/ D c0 C c1 x; x 2 I; c0 ; c1 2 Rg
(1.1)
In other words P1 .I / contains all functions of the form v.x/ D c0 C c1 x on I . Perhaps the most natural basis for P1 .I / is the monomial basis f1; xg, since any function v in P1 .I / can be written as a linear combination of 1 and x. That is, a constant c0 times 1 plus another constant c1 times x. In doing so, v is clearly determined by specifying c0 and c1 , the so-called coefficients of the linear combination. Indeed, we say that v has two degrees of freedom. However, c0 and c1 are not the only degrees of freedom possible for v. To see this, recall that a line, or linear function, is uniquely determined by requiring it to pass through any two given points. Now, obviously, there are many pairs of points that can specify the same line. For example, .0; 1/ and .2; 3/ can be used to specify v D x C1, but so can .1; 0/ and .4; 5/. In fact, any pair of points within the interval I will do as degrees of freedom for v. In particular, v can be uniquely determined by its values ˛0 D v.x0 / and ˛1 D v.x1 / at the end-points x0 and x1 of I . M.G. Larson and F. Bengzon, The Finite Element Method: Theory, Implementation, and Applications, Texts in Computational Science and Engineering 10, DOI 10.1007/978-3-642-33287-6__1, © Springer-Verlag Berlin Heidelberg 2013
1
2
1 Piecewise Polynomial Approximation in 1D
To prove this, let us assume that the values ˛0 D v.x0 / and ˛1 D v.x1 / are given. Inserting x D x0 and x D x1 into v.x/ D c0 C c1 x we obtain the linear system ˛ 1 x0 c0 D 0 (1.2) 1 x1 c1 ˛1 for ci , i D 1; 2. Computing the determinant of the system matrix we find that it equals x1 x0 , which also happens to be the length of the interval I . Hence, the determinant is positive, and therefore there exist a unique solution to (1.2) for any right hand side vector. Moreover, as a consequence, there is exactly one function v in P1 .I /, which has the values ˛0 and ˛1 at x0 and x1 , respectively. In the following we shall refer to the points x0 and x1 as the nodes. We remark that the system matrix above is called a Vandermonde matrix. Knowing that we can completely specify any function in P1 .I / by its node values ˛0 and ˛1 we now introduce a new basis f0 ; 1 g for P1 .I /. This new basis is called a nodal basis, and is defined by ( 1; if i D j j .xi / D ; i; j D 0; 1 (1.3) 0; if i ¤ j From this definition we see that each basis funct
Data Loading...
