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Peter Monk - One of the best experts on this subject based on the ideXlab platform.

  • Error estimates for the Ultra Weak Variational Formulation of the Helmholtz Equation
    ESAIM: Mathematical Modelling and Numerical Analysis, 2008
    Co-Authors: Annalisa Buffa, Peter Monk
    Abstract:

    The Ultra Weak Variational Formulation (UWVF) of the Helmholtz Equation provides a variational framework suitable for discretization using plane wave solutions of an appropriate adjoint Equation. Currently convergence of the method is only proved on the boundary of the domain. However substantial computational evidence exists showing that the method also converges throughout the domain of the Helmholtz Equation. In this paper we exploit the fact that the UWVF is essentially an upwind discontinuous Galerkin method to prove convergence of the solution in the special case where there is no absorbing medium present. We also provide some other estimates in the case when absorption is present, and give some simple numerical results to test the estimates. We expect that similar techniques can be used to prove error estimates for the UWVF applied to Maxwell's Equations and elasticity.

  • The perfectly matched layer for the ultra weak variational formulation of the 3D Helmholtz Equation
    International Journal for Numerical Methods in Engineering, 2004
    Co-Authors: Tomi Huttunen, Jari P. Kaipio, Peter Monk
    Abstract:

    We investigate the feasibility of using the perfectly matched layer (PML) as an absorbing boundary condition for the ultra weak variational formulation (UWVF) of the 3D Helmholtz Equation. The PML is derived using complex stretching of the spatial variables. This leads to a modified Helmholtz Equation for which the UWVF can be derived. In the standard discrete UWVF, the approximating subspace is constructed from local solutions of the Helmholtz Equation. In previous studies plane wave basis functions have been advocated because they simplify the building of the UWVF matrices. For the PML domain we propose a special set of plane wave basis functions which allow fast computations and efficiently reduce spurious numerical reflections. The method is validated by numerical experiments. In comparison to a low-order absorbing boundary condition, the PML shows superior performance. Copyright © 2004 John Wiley & Sons, Ltd.

  • a least squares method for the Helmholtz Equation
    Computer Methods in Applied Mechanics and Engineering, 1999
    Co-Authors: Peter Monk, Daqing Wang
    Abstract:

    Abstract We investigate the use of least-squares methods to approximate the Helmholtz Equation. The basis used in the discrete method consists of solutions of the Helmholtz Equation (either consisting of plane waves or Bessel functions) on each element of a finite element grid. Unlike previous methods of this type, we do not use polynomial based finite elements. The use of small elements (and relatively few basis functions per element) allows us to prove convergence theorems for the method and, to some extent, control the conditioning of the resulting linear sy stem. Numerical results show the efficiency of the new method and suggest that it may be possible to obtain accurate results with a coarser grid than is usual for standard finite element methods.

Eli Turkel - One of the best experts on this subject based on the ideXlab platform.

  • iterative schemes for high order compact discretizations to the exterior Helmholtz Equation
    Mathematical Modelling and Numerical Analysis, 2012
    Co-Authors: Yogi A Erlangga, Eli Turkel
    Abstract:

    We consider high order finite difference approximations to the Helmholtz Equation in an exterior domain. We include a simplified absorbing boundary condition to approximate the Sommerfeld radiation condition. This yields a large, but sparse, complex system, which is not self-adjoint and not positive definite. We discretize the Equation with a compact fourth or sixth order accurate scheme. We solve this large system of linear Equations with a Krylov subspace iterative method. Since the method converges slowly, a preconditioner is introduced, which is a Helmholtz Equation but with a modified complex wavenumber. This is discretized by a second or fourth order compact scheme. The system is solved by BICGSTAB with multigrid used for the preconditioner. We study, both by Fourier analysis and computations this preconditioned system especially for the effects of high order discretizations.

  • high order finite difference methods for the Helmholtz Equation
    Computer Methods in Applied Mechanics and Engineering, 1998
    Co-Authors: I Singer, Eli Turkel
    Abstract:

    High-order finite difference methods for solving the Helmholtz Equation are developed and analyzed, in one and two dimensions on uniform grids. The standard pointwise representation has a second-order accurate local truncation error. We also study two schemes which have a fourth-order accurate local truncation error. One of the high-order schemes is based on generalizations of the Pade approximation. The second scheme is based on high-order approximation to the derivative calculated from the Helmholtz Equation itself. A symmetric high-order representation is developed for a Neumann boundary condition. Numerical results are presented on model problems approximated with the developed schemes.

Yogi A Erlangga - One of the best experts on this subject based on the ideXlab platform.

Lexing Ying - One of the best experts on this subject based on the ideXlab platform.

  • sweeping preconditioner for the Helmholtz Equation moving perfectly matched layers
    Multiscale Modeling & Simulation, 2011
    Co-Authors: Bjorn Engquist, Lexing Ying
    Abstract:

    This paper introduces a new sweeping preconditioner for the iterative solution of the variable coefficient Helmholtz Equation in two and three dimensions. The algorithms follow the general structur ...

  • sweeping preconditioner for the Helmholtz Equation hierarchical matrix representation
    Communications on Pure and Applied Mathematics, 2011
    Co-Authors: Bjorn Engquist, Lexing Ying
    Abstract:

    The paper introduces the sweeping preconditioner, which is highly efficient for iterative solutions of the variable-coefficient Helmholtz Equation including very-high-frequency problems. The first central idea of this novel approach is to construct an approximate factorization of the discretized Helmholtz Equation by sweeping the domain layer by layer, starting from an absorbing layer or boundary condition. Given this specific order of factorization, the second central idea is to represent the intermediate matrices in the hierarchical matrix framework. In two dimensions, both the construction and the application of the preconditioners are of linear complexity. The generalized minimal residual method (GMRES) solver with the resulting preconditioner converges in an amazingly small number of iterations, which is essentially independent of the number of unknowns. This approach is also extended to the three-dimensional case with some success. Numerical results are provided in both two and three dimensions to demonstrate the efficiency of this new approach.

  • sweeping preconditioner for the Helmholtz Equation hierarchical matrix representation
    arXiv: Numerical Analysis, 2010
    Co-Authors: Bjorn Engquist, Lexing Ying
    Abstract:

    The paper introduces the sweeping preconditioner, which is highly efficient for iterative solutions of the variable coefficient Helmholtz Equation including very high frequency problems. The first central idea of this novel approach is to construct an approximate factorization of the discretized Helmholtz Equation by sweeping the domain layer by layer, starting from an absorbing layer or boundary condition. Given this specific order of factorization, the second central idea of this approach is to represent the intermediate matrices in the hierarchical matrix framework. In two dimensions, both the construction and the application of the preconditioners are of linear complexity. The GMRES solver with the resulting preconditioner converges in an amazingly small number of iterations, which is essentially independent of the number of unknowns. This approach is also extended to the three dimensional case with some success. Numerical results are provided in both two and three dimensions to demonstrate the efficiency of this new approach.

Stefan A Sauter - One of the best experts on this subject based on the ideXlab platform.

  • stability and finite element error analysis for the Helmholtz Equation with variable coefficients
    Mathematics of Computation, 2019
    Co-Authors: Ivan G Graham, Stefan A Sauter
    Abstract:

    We discuss the stability theory and numerical analysis of the Helmholtz Equation with variable and possibly nonsmooth or oscillatory coefficients. Using the unique continuation principle and the Fredholm alternative, we first give an existence-uniqueness result for this problem, which holds under rather general conditions on the coefficients and on the domain. Under additional assumptions, we derive estimates for the stability constant (i.e., the norm of the solution operator) in terms of the data (i.e., PDE coefficients and frequency), and we apply these estimates to obtain a new finite element error analysis for the Helmholtz Equation which is valid at a high frequency and with variable wave speed. The central role played by the stability constant in this theory leads us to investigate its behaviour with respect to coefficient variation in detail. We give, via a 1D analysis, an a priori bound with the stability constant growing exponentially in the variance of the coefficients (wave speed and/or diffusion coefficient). Then, by means of a family of analytic examples (supplemented by numerical experiments), we show that this estimate is sharp.

  • wavenumber explicit convergence analysis for galerkin discretizations of the Helmholtz Equation
    SIAM Journal on Numerical Analysis, 2011
    Co-Authors: Jens Markus Melenk, Stefan A Sauter
    Abstract:

    We develop a stability and convergence theory for a class of highly indefinite elliptic boundary value problems (bvps) by considering the Helmholtz Equation at high wavenumber $k$ as our model problem. The key element in this theory is a novel $k$-explicit regularity theory for Helmholtz bvps that is based on decomposing the solution into two parts: the first part has the Sobolev regularity properties expected of second order elliptic PDEs but features $k$-independent regularity constants; the second part is an analytic function for which $k$-explicit bounds for all derivatives are given. This decomposition is worked out in detail for several types of bvps, namely, the Helmholtz Equation in bounded smooth domains or convex polygonal domains with Robin boundary conditions and in exterior domains with Dirichlet boundary conditions. We present an error analysis for the classical $hp$-version of the finite element method ($hp$-FEM) where the dependence on the mesh width $h$, the approximation order $p$, and the wavenumber $k$ is given explicitly. In particular, under the assumption that the solution operator for Helmholtz problems is polynomially bounded in $k$, it is shown that quasi optimality is obtained under the conditions that $kh/p$ is sufficiently small and the polynomial degree $p$ is at least O(log $k$).

  • a generalized finite element method for solving the Helmholtz Equation in two dimensions with minimal pollution
    Computer Methods in Applied Mechanics and Engineering, 1995
    Co-Authors: Ivo Babuska, Frank Ihlenburg, Ellen T Paik, Stefan A Sauter
    Abstract:

    When using the Galerkin FEM for solving the Helmholtz Equation in two dimensions, the error of the corresponding solution differs substantially from the error of the best approximation, and this effect increases with higher wave number k. In this paper we will design a Generalized Finite Element Method (GFEM) for the Helmholtz Equation such that the pollution effect is minimal.