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\title{B-Spline Approximation of Neumann 
Problems}

\author{Klaus H\"ollig, Ulrich Reif, 
and Joachim Wipper}
\maketitle

\begin{abstract}
We describe a new finite element method for 
Neumann problems using cardinal splines.
The essential idea is the construction of a 
stable basis, so that B-splines with very small 
support in the domain do not lead to an
excessively large condition number of the 
Galerkin matrix.
This simple approach does not require any grid 
generation and yields smooth, high order 
accurate approximations with relatively low 
dimensional subspaces.
\end{abstract}

%%%%%

\section{Introduction}

One of the advantages of finite elements 
is the flexibility in the choice of the 
approximation methods.
If we denote by $B_k$ the basis functions of 
a finite element subspace $\mathbb{B}_h$ with 
mesh width $h$, the following, fairly general 
conditions are sufficient to guarantee 
stability and convergence for solving standard 
second order elliptic problems.

\medskip\noindent 
{\bf Local Support:} 
$|\text{supp}\,B_k|\preceq h$. 
\hfill $({\text{{\bf B}}}_{\text{{\bf L}}})$

\medskip\noindent 
{\bf Normalization:} 
$\|B_k\|_0\preceq 1$,
$\|B_k\|_1\preceq h^{-1}$.
\hfill $({\text{{\bf B}}}_{\text{{\bf N}}})$

\medskip\noindent 
{\bf Stability:} 
$|a_i|\preceq 
\|\sum_k a_kB_k\|_{0,\text{supp}\,B_i}$. 
\hfill $({\text{{\bf B}}}_{\text{{\bf S}}})$

\medskip\noindent 
{\bf Accuracy:} 
$\mathbb{B}_h$ contains polynomials of order 
$n$.
\hfill $({\text{{\bf B}}}_{\text{{\bf A}}})$

\medskip\noindent
Here, $|Q|$ denotes the diameter of a set 
$Q$ and
\[
\|v\|_\ell = \|v\|_{\ell,\Omega} = 
\left( \sum_{|\alpha|\le\ell} 
\int_\Omega |D^\alpha v|^2 \right)^{1/2}
\]
the Sobolev norm of a function $v$ on a domain 
$\Omega\subset\mathbb{R}^m$.
Moreover, we write 
\[
a \preceq b, 
\]
if $a\le c b$ with a constant $c$, which does 
not depend on the mesh width $h$.
Finally, we denote later on by $\|A\|$ the 
$2$-norm of a vector $A$.


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