corner treatments for high order local absorbing boundary conditions in high frequency acoustic scattering this paper deals with the design and validation of accurate local absorbing boundary conditions set on convex polygonal computational domains for the finite element solution of high frequency acoustic scattering problems while high order absorbing boundary conditions habcs are accurate for smooth fictitious boundaries the precision of the solution drops in the presence of corners if no specific treatment is applied we present and analyze two strategies to preserve the accuracy of pade type habcs at corners first by using compatibility relations derived for right angle corners and second by regularizing the boundary at the corner we show that the former strategy is well adapted to right corners and efficient for nearly right corners while the later is better for very obtuse corners numerical results are proposed to analyze and compare the approaches for two and three dimensional problems

absorbing boundary conditions for difference approximations to the multi dimensional wave equation construction des conditions aux limites absorbantes pour l equation des ondes multidimensionnelle a l aide d approximations aux differences formulation sous forme canonique des conditions aux limites absorbantes analytiques

optimized schwarz methods without overlap for the helmholtz equation the classical schwarz method is a domain decomposition method to solve elliptic partial differential equations in parallel convergence is achieved through overlap of the subdomains we study in this paper a variant of the schwarz method which converges without overlap for the helmholtz equation we show that the key ingredients for such an algorithm are the transmission conditions we derive optimal transmission conditions which lead to convergence of the algorithm in a finite number of steps these conditions are however nonlocal in nature and we introduce local approximations which we optimize for performance of the schwarz method this leads to an algorithm in the class of optimized schwarz methods we present an asymptotic analysis of the optimized schwarz method for two types of transmission conditions robin conditions and transmission conditions with second order tangential derivatives numerical results illustrate the effectiveness of the optimized schwarz method on a model problem and on a problem from industry

fictitious domain methods for the numerical solution of two dimensional scattering problems fictitious domain methods for the numerical solution of two dimensional scattering problems are considered the original exterior boundary value problem is approximated by truncating the unbounded domain and by imposing a nonreflecting boundary condition on the artificial boundary first order second order and exact nonreflecting boundary conditions are tested on rectangular and circular boundaries the finite element discretizations of the corresponding approximate boundary value problems are performed using locally fitted meshes and the discrete equations are solved with fictitious domain methods a special finite element method using nonmatching meshes is considered this method uses the macro hybrid formulation based on domain decomposition to couple polar and cartesian coordinate systems a special preconditioner based on fictitious domains is introduced for the arising algebraic saddle point system such that the subspace of constraints becomes invariant with respect to the preconditioned iterative procedure the performance of the new method is compared to the fictitious domain methods both with respect to accuracy and computational cost

fast direct solution of the helmholtz equation with a perfectly matched layer or an absorbing boundary condition we consider the efficient numerical solution of the helmholtz equation in a rectangular domain with a perfectly matched layer pml or an absorbing boundary condition abc standard bilinear trilinear finite element discretization on an orthogonal mesh leads to a separable system of linear equations for which we describe a cyclic reduction type fast direct solver we present numerical studies to estimate the reflection of waves caused by an absorbing boundary and a pml and we optimize certain parameters of the layer to minimize the reflection copyright 2003 john wiley sons ltd

second order absorbing boundary conditions for the wave equation a solution for the corner problem the treatment of domains with corners in the problem of absorbing boundary conditions for the wave equation is very important from a practical point of view a technical difficulty appears as soon as conditions of order greater than or equal to 2 are considered a solution is proposed for the two dimensional case when second order conditions are used this solution consists of prescribing an adequate corner condition the problem thus obtained is analyzed theoretically and the condition is proved to be optimal the results obtained here are illustrated by numerical simulations some extensions to higher space dimensions and higher order conditions are proposed

a damping preconditioner for time harmonic wave equations in fluid and elastic material a physical damping is considered as a preconditioning technique for acoustic and elastic wave scattering the earlier preconditioners for the helmholtz equation are generalized for elastic materials and three dimensional domains an algebraic multigrid method is used in approximating the inverse of damped operators several numerical experiments demonstrate the behavior of the method in complicated two dimensional and three dimensional domains

a parallel fictitious domain method for the three dimensional helmholtz equation the application of a fictitious domain domain embedding method to the three dimensional helmholtz equation with absorbing boundary conditions is considered the finite element discretization is performed by using locally fitted meshes and an algebraic fictitious domain method with a separable preconditioner is used in the iterative solution of the resultant linear systems such a method is based on embedding the original domain into a larger one with a simple geometry with this approach it is possible to realize the gmres iterations in a low dimensional subspace and use the partial solution method to solve the linear systems with the preconditioner an efficient parallel implementation of the iterative algorithm is introduced results of numerical experiments demonstrate good scalability properties on distributed memory parallel computers and the ability to solve high frequency acoustic scattering problems

equation based interpolation and incremental unknowns for solving the three dimensional helmholtz equation abstract in an earlier paper poullet and boag 2007 1 we developed an efficient incremental unknowns iu preconditioner for solving the two dimensional 2d helmholtz problem in both high and low frequency wavenumber regimes the multilevel preconditioning scheme involves separation of each grid into a coarser grid of the following level and a complementary grid on which the ius are defined by interpolation this approach is efficient as long as the mesh size of the coarsest grid is sufficiently small compared to the wavelength in order to overcome this restriction the authors introduced recently in poullet and boag 2010 2 a modified iu method combining the conventional interpolation with the helmholtz equation based interpolation ebi the ebi coefficients are derived numerically using a sufficiently large set of analytic solutions of the helmholtz equation on a special hierarchy of stencils the modified ius using helmholtz ebi are shown to provide improved preconditioning on the coarse scales where the conventional interpolation can not be employed this study deals with the extension of this idea for solving the three dimensional 3d helmholtz equation

a domain decomposition solver for acoustic scattering by elastic objects in layered media a finite element solution procedure is presented for accurately computing time harmonic acoustic scattering by elastic targets buried in sediment an improved finite element discretization based on trilinear basis functions leading to fourth order phase accuracy is described for sufficiently accurate discretizations 100 million to 1 billion unknowns are required the resulting systems of linear equations are solved iteratively using the gmres method with a domain decomposition preconditioner employing a fast direct solver due to the construction of the discretization and preconditioner iterations can be reduced onto a sparse subspace associated with the interfaces numerical experiments demonstrate capability to evaluate the scattered field with hundreds of wavelengths

continued fraction absorbing boundary conditions for convex polygonal domains continued fraction absorbing boundary conditions cfabcs are highly effective boundary conditions for modelling wave absorption into unbounded domains they are based on rational approximation of the exact dispersion relationship and were originally developed for straight computational boundaries in this paper cfabcs are extended to the more general case of polygonal computational domains the key to the current development is the surprising link found between the cfabcs and the complex co ordinate stretching of perfectly matched layers pmls this link facilitates the extension of cfabcs to oblique corners and thus to polygonal domains it is shown that the proposed cfabcs are easy to implement expected to perform better than pmls and are effective for general polygonal computational domains in addition to the derivation of cfabcs a novel explicit time stepping scheme is developed for efficient numerical implementation numerical examples presented in the paper illustrate that effective absorption is attained with a negligible increase in the computational cost for the interior domain although this paper focuses on wave propagation its theoretical development can be easily extended to the more general class of problems where the governing differential equation is second order in space with constant coefficients copyright 2005 john wiley sons ltd

a parallel multigrid based preconditioner for the 3d heterogeneous high frequency helmholtz equation we investigate the parallel performance of an iterative solver for 3d heterogeneous helmholtz problems related to applications in seismic wave propagation for large 3d problems the computation is no longer feasible on a single processor and the memory requirements increase rapidly therefore parallelization of the solver is needed we employ a complex shifted laplace preconditioner combined with the bi cgstab iterative method and use a multigrid method to approximate the inverse of the resulting preconditioning operator a 3d multigrid method with 2d semi coarsening is employed we show numerical results for large problems arising in geophysical applications

absorbing boundary condition for scalar wave propagation problems in infinite media based on a root finding algorithm abstract in this study we propose a new approach for development of absorbing boundary conditions for scalar wave propagation problems in infinite media based on a root finding algorithm for the solution of the exact wave dispersion relation we select the newton raphson method as the root finding algorithm in the present study and assess the accuracy of the newly developed boundary condition by estimating its reflection coefficient furthermore we evaluate and verify the stability of the boundary condition we apply our development to various scalar wave propagation problems and demonstrate that the proposed approach leads to accurate and stable computations

an algebraic multigrid based shifted laplacian preconditioner for the helmholtz equation a preconditioner defined by an algebraic multigrid cycle for a damped helmholtz operator is proposed for the helmholtz equation this approach is well suited for acoustic scattering problems in complicated computational domains and with varying material properties the spectral properties of the preconditioned systems and the convergence of the gmres method are studied with linear quadratic and cubic finite element discretizations numerical experiments are performed with two dimensional problems describing acoustic scattering in a cross section of a car cabin and in a layered medium asymptotically the number of iterations grows linearly with respect to the frequency while for lower frequencies the growth is milder the proposed preconditioner is particularly effective for low frequency and mid frequency problems

iterative solution of 3d acoustic wave equation with perfectly matched layer boundary condition and multigrid preconditioner abstractwe tested a biconjugate gradient stabilized bicgstab solver using a multigrid based preconditioner for solving the acoustic wave helmholtz equation in the frequency domain the perfectly matched layer pml method was used as the radiation boundary condition rbc the equation was discretized using either a second or fourth order finite difference fd scheme the convergence of an iterative solver depended strongly on the rbc used because the spectrum of the discretized equation also depends on it we used a geometric multigrid approach to construct a preconditioner for our fd frequency domain fdfd forward solver equipped with the pml boundary condition for efficiency this preconditioner was only constructed using a second order fd scheme with negligible attenuation inside the pml domain the preconditioner was used for accelerating the convergence rate of the fdfd forward solver for cases when the discretization grids were oversampled i e when the number of discretization points per

riesz operators with finite rank iterates abstract every infinite dimensional banach space admits riesz operators that are not finite rank in this note we discuss conditions under which a riesz operator or some power thereof is a finite rank operator

migration based traveltime waveform inversion of 2 d simple structures a synthetic example migration based traveltime mbtt formulation provides algorithms for automatically determining background velocities from full waveform surface seismic reflection data using local optimization methods in particular it addresses the difficulty of the nonconvexity of the least squares data misfit function the method consists of parameterizing the reflectivity in the time domain through a migration step and providing a multiscale representation for the smooth background velocity we present an implementation of the mbtt approach for a 2 d finite difference fd full wave acoustic model numerical analysis on a 2 d synthetic example shows the ability of the method to find much more reliable estimates of both long and short wavelengths of the velocity than the classical least squares approach even when starting from very poor initial guesses this enlargement of the domain of attraction for the global minima of the least squares misfit has a price each evaluation of the new objective function requires besides the usual fd full wave forward modeling an additional full wave prestack migration hence the fd implementation of the mbtt approach presented in this paper is expected to provide a useful tool for the inversion of data sets of moderate size

a multigrid method enhanced by krylov subspace iteration for discrete helmholtz equations standard multigrid algorithms have proven ineffective for the solution of discretizations of helmholtz equations in this work we modify the standard algorithm by adding gmres iterations at coarse levels and as an outer iteration we demonstrate the algorithm s effectiveness through theoretical analysis of a model problem and experimental results in particular we show that the combined use of gmres as a smoother and outer iteration produces an algorithm whose performance depends relatively mildly on wave number and is robust for normalized wave numbers as large as 200 for fixed wave numbers it displays grid independent convergence rates and has costs proportional to the number of unknowns

microlocal approach towards construction of nonreflecting boundary conditions abstract this paper addresses the problem of construction of non reflecting boundary condition for certain second order nonlinear dispersive equations it is shown that using the concept of microlocality it is possible to relax the requirement of compact support of the initial data the method is demonstrated for a class of initial data such that outside the computational domain it behaves like a continuous wave the generalization is detailed for two existing schemes in the framework of pseudo differential calculus namely szeftel s method szeftel 2006 1 and gauge transformation strategy antoine et al 2006 2 efficient numerical implementation is discussed and a comparative performance analysis is presented the paper also briefly surveys the possibility of extension of the method to higher dimensional pdes

versal deformations and normal forms for reversible and hamiltonian linear systems the problem of this article is the characterization of equivalence classes and their versal deformations for reversible and reversible hamiltonian matrices in both cases the admissible transformations form a subgroup g of gl m therefore the gl m orbits of a given matrix may split into several g orbits these orbits are characterized by signs for each sign we have a normal form and a corresponding versal deformation the main tool in the characterization is reduction to the semi simple case c 1996 academic press inc

dirichlet to neumann map for domains with corners and approximate boundary conditions we present in this paper the dirichlet to neumann operator for the wave equation on a straight wedge in r 2 using fourier integral operators as a consequence we recover the classical approximate boundary conditions of orders 1 and 2

absorbing boundary conditions for linear gravity waves this article constructs analyzes and implements a family of absorbing boundary conditions for linear gravity waves in two dimensions the main difficulty consists in considering the dispersive nature of these waves

optimal discontinuous galerkin methods for the acoustic wave equation in higher dimensions in this paper we develop and analyze a new class of discontinuous galerkin dg methods for the acoustic wave equation in mixed form traditional mixed finite element fe methods produce energy conserving schemes but these schemes are implicit making the time stepping inefficient standard dg methods give explicit schemes but these approaches are typically dissipative or suboptimally convergent depending on the choice of numerical fluxes our new method can be seen as a compromise between these two kinds of techniques in the way that it is both explicit and energy conserving locally and globally moreover it can be seen as a generalized version of the raviart thomas fe method and the finite volume method stability and convergence of the new method are rigorously analyzed and we have shown that the method is optimally convergent furthermore in order to apply the new method for unbounded domains we apply our new method with the first order absorbing boundary condition the stability of the resulting numerical scheme is analyzed