Читать книгу Finite Element Analysis - Barna Szabó - Страница 23

The standard polynomial space of degree p, denoted by , is spanned by the monomials defined on the standard element

(1.49)

The choice of basis functions is guided by considerations of implementation, keeping the condition number of the coefficient matrices small, and personal preferences. For the symmetric positive‐definite matrices considered here the condition number C is the largest eigenvalue divided by the smallest. The number of digits lost in solving a linear problem is roughly equal to . Characterizing the condition number as being large or small should be understood in this context. In the finite element method the condition number depends on the choice of the basis functions and the mesh.

The standard polynomial basis functions, called shape functions, can be defined in various ways. We will consider shape functions based on Lagrange polynomials and Legendre¹² polynomials. We will use the same notation for both types of shape function.