Piecewise polynomial basis

Silverstone and his collaborators have proposed the use of piecewise polynomial basis functions and have demonstrated their use for atomic systems. By using such functions, they aim to overcome the computational linear dependence problems which may be associated with large basis sets of exponential-type functions or Gaussian-type functions. 

The integrals over piecewise polynomial basis functions are elementary. However, it is found that the number of terms in the Hnal integral formulae is much larger for piecewise polynomial functions than for exponential-type functions, for example. 

The set of B-splines of order k on the knot sequence f< forms a complete basis for piecewise polynomials of degree fc — 1 on the interval spanned by the knot sequence. We represent the solution to the radial Dirac equation as a linear combination of these B-splines and work with the B-spline representation of the wave functions rather than the wave functions themselves. 

Unlike the traditional Taylor s series expansion method, the Galerkin approach utilizes basis functions, such as linear piecewise polynomials, to approximate the true solution. For example, the Galerkin approximation to the sample problem Equation 23.1 would require evaluating Equation 23.13 for the specific grid formation and specific choice of basis function ... 

One modified version of the spectral method is called the finite-element method. The basis functions fij of Eq. (37) in the spectral method are defined over the entire domain, but the basis functions of the finite-element method are piecewise polynomials. These polynomials are local so that they are nonzero over only a small finite element. For example, piecewise linear roof functions have been used as basis functions. Coefficients Cj t) are then determined by appUcation of the Galeikin approximation. Because the basis functions are nonzero over only a small domain, the expression for W(x, t) resembles a finite-difference form. 

The present paper follows our two earlier contributions, Coimbra (2000 and 2002) where we have presented the formal treatment of moving finite element method with a piecewise higher degree polynomial basis in space. Without loss of generality we will only describe the MFEM in 2D. The mathematical model of a process involving diffusion, reaction and convections in 2D usually consists of an equation of the form... 

Up to now all the computational procedures have been precisely defined, with the exception of the basis set. In the atomic case all the equations are of course simplified to only the radial coordinate, the angular part being analytical. The radial functions are expanded in a basis set of B-splines over a selected interval [0, RmaxI- These are piecewise polynomial functions, completely defined in terms of... 

Equation (1), with the associated boundary conditions, is a nonlinear second-order boundary-value ODE. This was solved by the method of collocation with piecewise cubic Hermite polynomial basis functions for spatial discretization, while simple successive substitution was adequate for the solution of the resulting nonlinear algebraic equations. The method has been extensively described before [9], and no problems were found in this application. 

An important feature of the method of lines is selection of the basis functions i (co), which determines the precision of (spatial) curve fitting. The piecewise polynomials known as B splines meet the requirements. Curve fitting by means of spline functions entails division of the solution space into subintervals by means of a series of points called knots. Knots may be either single or multiple, a multiple knot being formed by the coincidence of two or more such points. They are numbered in nondecreasing order of location Si, S2,..., 5i,. A normalized B spline of order k takes nonzero values only over a range of k subintervals between knots, and, for example, Bij (co), the ith normalized B spline of order k for the knot sequence s, is zero outside the interval + nonnegative at = s, and w = Si + j, and strictly... 

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