In this paper, we study the electromagnetic scattering from a two dimensional large rectangular open cavity embedded in an infinite ground plane, which is
modelled by Helmholtz equations. By introducing nonlocal transparent boundary conditions, the problem in the open cavity is reduced to a bounded domain problem. A
hypersingular integral operator and a weakly singular integral operator are involved in
the TM and TE cases, respectively. A new second-order Toeplitz type approximation
and a second-order finite difference scheme are proposed for approximating the hypersingular integral operator on the aperture and the Helmholtz in the cavity, respectively.
The existence and uniqueness of the numerical solution in the TE case are established
for arbitrary wavenumbers. A fast algorithm for the second-order approximation is proposed for solving the cavity model with layered media. Numerical results show the
second-order accuracy and efficiency of the fast algorithm. More important is that the
algorithm is easy to implement as a preconditioner for cavity models with more general
A class of normal-like derivatives for functions with low regularity defined
on Lipschitz domains are introduced and studied. It is shown that the new normal-like
derivatives, which are called the generalized normal derivatives, preserve the major properties of the existing standard normal derivatives. The generalized normal derivatives
are then applied to analyze the convergence of domain decomposition methods (DDMs)
with nonmatching grids and discontinuous Galerkin (DG) methods for second-order elliptic problems. The approximate solutions generated by these methods still possess
the optimal energy-norm error estimates, even if the exact solutions to the underlying
elliptic problems admit very low regularities.
Solution-driven mesh adaptation is becoming quite popular for spatial error
control in the numerical simulation of complex computational physics applications, such
as climate modeling. Typically, spatial adaptation is achieved by element subdivision ($h$ adaptation) with a primary goal of resolving the local length scales of interest. A second, less-popular method of spatial adaptivity is called "mesh motion" ($r$ adaptation);
the smooth repositioning of mesh node points aimed at resizing existing elements to
capture the local length scales. This paper proposes an adaptation method based on a
combination of both element subdivision and node point repositioning ($rh$ adaptation).
By combining these two methods using the notion of a mobility function, the proposed
approach seeks to increase the flexibility and extensibility of mesh motion algorithms
while providing a somewhat smoother transition between refined regions than is produced by element subdivision alone. Further, in an attempt to support the requirements
of a very general class of climate simulation applications, the proposed method is designed to accommodate unstructured, polygonal mesh topologies in addition to the most
popular mesh types.
The discontinuous Galerkin (DG) or local discontinuous Galerkin (LDG)
method is a spatial discretization procedure for convection-diffusion equations, which
employs useful features from high resolution finite volume schemes, such as the exact
or approximate Riemann solvers serving as numerical fluxes and limiters. The Lax-Wendroff time discretization procedure is an alternative method for time discretization
to the popular total variation diminishing (TVD) Runge-Kutta time discretizations. In
this paper, we develop fluxes for the method of DG with Lax-Wendroff time discretization procedure (LWDG) based on different numerical fluxes for finite volume or finite
difference schemes, including the first-order monotone fluxes such as the Lax-Friedrichs
flux, Godunov flux, the Engquist-Osher flux etc. and the second-order TVD fluxes. We
systematically investigate the performance of the LWDG methods based on these different numerical fluxes for convection terms with the objective of obtaining better performance by choosing suitable numerical fluxes. The detailed numerical study is mainly
performed for the one-dimensional system case, addressing the issues of CPU cost, accuracy, non-oscillatory property, and resolution of discontinuities. Numerical tests are
also performed for two dimensional systems.
Stochastic approximation problem is to find some roots or extremum of a nonlinear function for which only noisy measurements of the function are available. The
classical algorithm for stochastic approximation problem is the Robbins-Monro (RM)
algorithm, which uses the noisy evaluation of the negative gradient direction as the
iterative direction. In order to accelerate the RM algorithm, this paper gives a frame
algorithm using adaptive iterative directions. At each iteration, the new algorithm goes
towards either the noisy evaluation of the negative gradient direction or some other
directions under some switch criteria. Two feasible choices of the criteria are proposed and two corresponding frame algorithms are formed. Different choices of the
directions under the same given switch criterion in the frame can also form different
algorithms. We also proposed the simultanous perturbation difference forms for the
two frame algorithms. The almost surely convergence of the new algorithms are all
established. The numerical experiments show that the new algorithms are promising.
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