Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Higher-order accurate diffuse-domain methods for partial differential
  equations with Dirichlet boundary conditions in complex, evolving geometries   The diffuse-domain, or smoothed boundary, method is an attractive approach
for solving partial differential equations in complex geometries because of its
simplicity and flexibility. In this method the complex geometry is embedded
into a larger, regular domain. The original PDE is reformulated using a
smoothed characteristic function of the complex domain and source terms are
introduced to approximate the boundary conditions. The reformulated equation,
which is independent of the dimension and domain geometry, can be solved by
standard numerical methods and the same solver can be used for any domain
geometry. A challenge is making the method higher-order accurate. For Dirichlet
boundary conditions, which we focus on here, current implementations
demonstrate a wide range in their accuracy but generally the methods yield at
best first order accuracy in $\epsilon$, the parameter that characterizes the
width of the region over which the characteristic function is smoothed.
Typically, $\epsilon\propto h$, the grid size. Here, we analyze the
diffuse-domain PDEs using matched asymptotic expansions and explain the
observed behaviors. Our analysis also identifies simple modifications to the
diffuse-domain PDEs that yield higher-order accuracy in $\epsilon$, e.g.,
$O(\epsilon^2)$ in the $L^2$ norm and $O(\epsilon^p)$ with $1.5\le p\le 2$ in
the $L^{\infty}$ norm. Our analytic results are confirmed numerically in
stationary and moving domains where the level set method is used to capture the
dynamics of the domain boundary and to construct the smoothed characteristic
function.

35
The categories of this paper are: mtrl-th
19
The categories of this paper are: eess
0
The categories of this paper are: hep-ph
22
The categories of this paper are: chao-dyn
12
The categories of this paper are: nucl-th
25
The categories of this paper are: patt-sol
9
The categories of this paper are: nlin
7
The categories of this paper are: hep-th
10
The categories of this paper are: q-bio
29
The categories of this paper are: funct-an
26
The categories of this paper are: bayes-an
37
The categories of this paper are: supr-con
5
The categories of this paper are: gr-qc
30
The categories of this paper are: q-alg
20
The categories of this paper are: acc-phys
28
The categories of this paper are: alg-geom