Complementary families of approximating polynomials with applications to finite element methods applied to differential equations of arbitrary even spatial order

Abstract

Complementary families of polynomials are introduced to generate $C^m$ finite element basis functions of order $p \geq 2m+2$ for arbitrary $m \ge 0$. One family consists of the Hermite splines that serve as the nodal basis functions by ensuring $C^m$ continuity across element boundaries. Explicit formulas for these splines for any $m \ge 0$ are presented on the canonical interval $[0,1]$. The second family is derived on the interval $[-1,1]$ from derivatives of order $m+1$ of the Legendre polynomials of degree $p-m-1$ multiplied by binomial powers of degree $m+1$ at -1 and 1, respectively. These polynomials, related to the ultraspherical polynomials, serve as the interior or bubble basis functions. A relationship between the two families of polynomials is demonstrated. For a particular $m$ and $p$, an interpolant is constructed using these basis pairs together with the roots of the related ultraspherical polynomial and the interval endpoints. A formula for the interpolation error that extends the results for $m=0$ and $m=1$ is given. To prove the formula extensions of the Lagrange interpolants are introduced. A superconvergence result along with the related asymptotic equivalence of the interpolant and finite element solution is proved in the linear case in $H^{m+1}$. Computational results demonstrate the theory for a model problem.