Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Numerical Approximation of Fractional Powers of Elliptic Operators   In this paper, we develop and study algorithms for approximately solving the
linear algebraic systems: $\mathcal{A}_h^\alpha u_h = f_h$, $ 0< \alpha <1$,
for $u_h, f_h \in V_h$ with $V_h$ a finite element approximation space. Such
problems arise in finite element or finite difference approximations of the
problem $ \mathcal{A}^\alpha u=f$ with $\mathcal{A}$, for example, coming from
a second order elliptic operator with homogeneous boundary conditions. The
algorithms are motivated by the recent method of Vabishchevich, 2015, that
relates the algebraic problem to a solution of a time-dependent initial value
problem on the interval $[0,1]$. Here we develop and study two time stepping
schemes based on diagonal Pad\'e approximation to $(1+x)^{-\alpha}$. The first
one uses geometrically graded meshes in order to compensate for the singular
behavior of the solution for $t$ close to $0$. The second algorithm uses
uniform time stepping but requires smoothness of the data $f_h$ in discrete
norms. For both methods, we estimate the error in terms of the number of time
steps, with the regularity of $f_h$ playing a major role for the second method.
Finally, we present numerical experiments for $\mathcal{A}_h$ coming from the
finite element approximations of second order elliptic boundary value problems
in one and two spatial dimensions.

14
The categories of this paper are: math-ph
10
The categories of this paper are: q-bio
0
The categories of this paper are: hep-ph
33
The categories of this paper are: plasm-ph
4
The categories of this paper are: cond-mat
5
The categories of this paper are: gr-qc
35
The categories of this paper are: mtrl-th
24
The categories of this paper are: dg-ga
3
The categories of this paper are: physics
30
The categories of this paper are: q-alg
8
The categories of this paper are: hep-ex
6
The categories of this paper are: astro-ph
7
The categories of this paper are: hep-th
11
The categories of this paper are: hep-lat
16
The categories of this paper are: stat
36
The categories of this paper are: cmp-lg