This paper is concerned with an interior penalty discontinuous Galerkin (IPDG) method
based on a flexible type of non-polynomial local approximation space
for the Helmholtz equation with varying wavenumber.
The local approximation space consists of multiple polynomial-modulated
phase functions which can be chosen according to the phase information of the solution.
We obtain some {approximation} properties for this space
and \textit{a prior} $L^2$ error estimates
for the \textit{h}-convergence of the IPDG method
using duality argument.
We also provide ample numerical examples to show that,
building phase information into the local spaces
often gives more accurate results comparing to using the
standard polynomial spaces.
We derive pointwise curvature estimates for graphical mean curvature flows in higher codimensions. To the best of our knowledge, this is the first such estimates without assuming smallness of first derivatives of the defining map. An immediate application is a convergence theorem of the mean curvature flow of the graph of an area decreasing map between flat Riemann surfaces.
Let Ω⊂$R$^{$n$}be an arbitrary open set. In this paper it is shown that if a Sobolev function$f$∈$W$^{1,$p$}(Ω) possesses a zero trace (in the sense of Lebesgue points) on ϖΩ, then$f$is weakly zero on ϖΩ in the sense that$f$∈$W$_{0}^{1,$p$}(Ω).