scholarly journals An absolutely stable $hp$-HDG method for the time-harmonic Maxwell equations with high wave number

2016 ◽  
Vol 86 (306) ◽  
pp. 1553-1577 ◽  
Author(s):  
Peipei Lu ◽  
Huangxin Chen ◽  
Weifeng Qiu
Keyword(s):  
Maxwell Equations ◽  
Wave Number ◽  
High Wave Number ◽  
High Wave ◽  
Time Harmonic

A Hybridizable Discontinuous Galerkin Method for the Time-Harmonic Maxwell Equations with High Wave Number

2016 ◽  
Vol 16 (3) ◽  
pp. 429-445 ◽  
Author(s):  
Xiaobing Feng ◽  
Peipei Lu ◽  
Xuejun Xu
Keyword(s):  
Discontinuous Galerkin ◽  
Maxwell Equations ◽  
Impedance Boundary Condition ◽  
Wave Number ◽  
High Wave Number ◽  
Impedance Boundary ◽  
High Wave ◽  
Hybridizable Discontinuous Galerkin ◽  
Time Harmonic ◽  
The Stability

AbstractThis paper proposes and analyzes a hybridizable discontinuous Galerkin (HDG) method for the three-dimensional time-harmonic Maxwell equations coupled with the impedance boundary condition in the case of high wave number. It is proved that the HDG method is absolutely stable for all wave numbers ${\kappa>0}$ in the sense that no mesh constraint is required for the stability. A wave-number-explicit stability constant is also obtained. This is done by choosing a specific penalty parameter and using a PDE duality argument. Utilizing the stability estimate and a non-standard technique, the error estimates in both the energy-norm and the ${\mathbf{L}^{2}}$-norm are obtained for the HDG method. Numerical experiments are provided to validate the theoretical results and to gauge the performance of the proposed HDG method.

The Method of Fundamental Solutions for Solving Exterior Axisymmetric Helmholtz Problems with High Wave-Number

2013 ◽  
Vol 5 (04) ◽  
pp. 477-493 ◽  
Author(s):  
Wen Chen ◽  
Ji Lin ◽  
C.S. Chen
Keyword(s):  
Large Scale ◽  
Three Dimensional ◽  
Matrix Decomposition ◽  
Coefficient Matrix ◽  
Fundamental Solutions ◽  
Method Of Fundamental Solutions ◽  
Wave Number ◽  
High Wave Number ◽  
Memory Space ◽  
High Wave

AbstractIn this paper, we investigate the method of fundamental solutions (MFS) for solving exterior Helmholtz problems with high wave-number in axisymmetric domains. Since the coefficient matrix in the linear system resulting from the MFS approximation has a block circulant structure, it can be solved by the matrix decomposition algorithm and fast Fourier transform for the fast computation of large-scale problems and meanwhile saving computer memory space. Several numerical examples are provided to demonstrate its applicability and efficacy in two and three dimensional domains.

scholarly journals Stable and Accurate Filtering Procedures

2020 ◽  
Vol 82 (1) ◽  
Author(s):  
Tomas Lundquist ◽  
Jan Nordström
Keyword(s):  
Boundary Conditions ◽  
High Frequency ◽  
Wave Number ◽  
Main Concern ◽  
High Wave Number ◽  
High Wave ◽  
Artificial Dissipation ◽  
High Frequency Oscillations ◽  
Energy Bound ◽  
Energy Stable

AbstractHigh frequency errors are always present in numerical simulations since no difference stencil is accurate in the vicinity of the $$\pi $$π-mode. To remove the defective high wave number information from the solution, artificial dissipation operators or filter operators may be applied. Since stability is our main concern, we are interested in schemes on summation-by-parts (SBP) form with weak imposition of boundary conditions. Artificial dissipation operators preserving the accuracy and energy stability of SBP schemes are available. However, for filtering procedures it was recently shown that stability problems may occur, even for originally energy stable (in the absence of filtering) SBP based schemes. More precisely, it was shown that even the sharpest possible energy bound becomes very weak as the number of filtrations grow. This suggest that successful filtering include a delicate balance between the need to remove high frequency oscillations (filter often) and the need to avoid possible growth (filter seldom). We will discuss this problem and propose a remedy.

scholarly journals A robust domain decomposition method for the Helmholtz equation with high wave number

2016 ◽  
Vol 50 (3) ◽  
pp. 921-944 ◽  
Author(s):  
Wenbin Chen ◽  
Yongxiang Liu ◽  
Xuejun Xu
Keyword(s):  
Helmholtz Equation ◽  
Domain Decomposition ◽  
Domain Decomposition Method ◽  
Mesh Size ◽  
Relaxation Parameter ◽  
Point Of View ◽  
Wave Number ◽  
Theoretical Point ◽  
High Wave Number ◽  
High Wave

In this paper we present a robust Robin−Robin domain decomposition (DD) method for the Helmholtz equation with high wave number. Through choosing suitable Robin parameters on different subdomains and introducing a new relaxation parameter, we prove that the new DD method is robust, which means the convergence rate is independent of the wave number k for kh = constant and the mesh size h for fixed k. To the best of our knowledge, from the theoretical point of view, this is a first attempt to design a robust DD method for the Helmholtz equation with high wave number in the literature. Numerical results which confirm our theory are given.

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