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Computationally Efficient Boundary Element Methods for High-Frequency Helmholtz Problems in Unbounded Domains [chapter]

Timo Betcke, Elwin van 't Wout, Pierre Gélat
2017 Modern Solvers for Helmholtz Problems  
This chapter presents the application of the boundary element method to high-frequency Helmholtz problems in unbounded domains.  ...  Based on a standard combined integral equation approach for sound-hard scattering problems we discuss the discretization, preconditioning and fast evaluation of the involved operators.  ...  For sufficiently smooth Γ this formulation is again a perturbation of a scaled identity because V κ D = (V + C)D, where C is a compact operator [12, Lemma 2.1] and V is the single-layer operator for the  ... 
doi:10.1007/978-3-319-28832-1_9 fatcat:mhsftqcgpvgklkn6jy3z7punru

Scalable implicit incompressible resistive MHD with stabilized FE and fully-coupled Newton–Krylov-AMG

J.N. Shadid, R.P. Pawlowski, E.C. Cyr, R.S. Tuminaro, L. Chacón, P.D. Weber
2016 Computer Methods in Applied Mechanics and Engineering  
Initial results that explore the scaling of the solution methods are also presented on up to 128K processors for problems with up to 1.8B unknowns on a CrayXK7.  ...  These preconditioners are shown to enable a robust, scalable and efficient solution approach for the large-scale sparse linear systems generated by the Newton linearization.  ...  The result of a fully-implicit (or a direct-to-steady-state) solution technique is the development of very large-scale, coupled highly nonlinear system(s) that must be solved.  ... 
doi:10.1016/j.cma.2016.01.019 fatcat:fr632t2xvjbvnfsapu6tox2ev4

Numerical-asymptotic boundary integral methods in high-frequency acoustic scattering

Simon N. Chandler-Wilde, Ivan G. Graham, Stephen Langdon, Euan A. Spence
2012 Acta Numerica  
These hybrid methods combine conventional piecewise polynomial approximations with high-frequency asymptotics to build basis functions suitable for representing the oscillatory solutions.  ...  In this article we describe recent progress on the design, analysis and implementation of hybrid numerical-asymptotic boundary integral methods for boundary value problems for the Helmholtz equation that  ...  Acknowledgements This review has benefited greatly from the contributions of our collaborators and also from the comments and suggestions of a number of people.  ... 
doi:10.1017/s0962492912000037 fatcat:xug2jil34rauthd2icxfocwkka

Book Review: Report on global methods for combinatorial isoperimetric problems

Igor Shparlinski
2005 Mathematics of Computation  
The additional errors incurred through the numerical approximation of the dual solution are difficult to quantify unless one embarks on reliable a posteriori error estimation for the dual problem; for  ...  R(u h ) and the error z − v h , with v h ∈ V h , in the approximation of the dual solution z, which acts as a weight function for the residual.  ...  Finally, a multigrid method is presented as a general concept for solving finite element systems.  ... 
doi:10.1090/s0025-5718-04-01757-0 fatcat:ddoei3mqenfhjoi5czaj5fbt3u

Stability and error analysis for the Helmholtz equation with variable coefficients [article]

I.G. Graham, S.A. Sauter
2019 arXiv   pre-print
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.  ...  We discuss the stability theory and numerical analysis of the Helmholtz equation with variable and possibly non-smooth or oscillatory coefficients.  ...  Acknowledgement We are grateful to the Hausdorff Research Institute for Mathematics in Bonn for Visiting Fellowships in their 2017 Trimester Programme on Multiscale Methods, during which part of this work  ... 
arXiv:1803.00966v2 fatcat:pfvaskam6vckxcrq43vzea7vpy

A fast and well-conditioned spectral method [article]

Sheehan Olver, Alex Townsend
2012 arXiv   pre-print
A spectral method is developed for the direct solution of linear ordinary differential equations with variable coefficients.  ...  We prove stability of the method by relating it to a diagonally preconditioned system which has a bounded condition number, in a suitable norm.  ...  Gonnet, discussions with whom led to the observation of well-conditioning of coefficient methods, which initiated the research of this paper. We also thank the rest of the Chebfun team, including T.  ... 
arXiv:1202.1347v2 fatcat:tea34oyqozbr3auurxdn7cdcoi

Computational Electromagnetism and Acoustics

Ralf Hiptmair, Ronald Hoppe, Patrick Joly, Ulrich Langer
2013 Oberwolfach Reports  
the focus of Stability Analysis of Time-Domain PML, whereas an analysis for periodic structures case was presented in On the far field of the solutions of Helmholtz equations in periodic waveguide.  ...  A profound analysis of polynomial Galerkin methods was given in hp-FEM and hp-DGFEM for Helmholtz problems, whereas DPG Method, an Overview.  ...  We solved integral equations (13) using the high-order Nyström methods introduced in [1] for a bisinusoidal grating whose height to period ratio equals to 1.  ... 
doi:10.4171/owr/2013/03 fatcat:znjejdppnnectpfwz35s5zmiai

Computational Electromagnetism and Acoustics

Ralf Hiptmair, Ronald Hoppe, Patrick Joly, Ulrich Langer
2007 Oberwolfach Reports  
the focus of Stability Analysis of Time-Domain PML, whereas an analysis for periodic structures case was presented in On the far field of the solutions of Helmholtz equations in periodic waveguide.  ...  A profound analysis of polynomial Galerkin methods was given in hp-FEM and hp-DGFEM for Helmholtz problems, whereas DPG Method, an Overview.  ...  We solved integral equations (13) using the high-order Nyström methods introduced in [1] for a bisinusoidal grating whose height to period ratio equals to 1.  ... 
doi:10.4171/owr/2007/05 fatcat:dx6fpblbjnbwtehcvi4b5iunym

Computational Electromagnetism and Acoustics

Ralf Hiptmair, Ronald Hoppe, Patrick Joly, Ulrich Langer
2010 Oberwolfach Reports  
the focus of Stability Analysis of Time-Domain PML, whereas an analysis for periodic structures case was presented in On the far field of the solutions of Helmholtz equations in periodic waveguide.  ...  A profound analysis of polynomial Galerkin methods was given in hp-FEM and hp-DGFEM for Helmholtz problems, whereas DPG Method, an Overview.  ...  We solved integral equations (13) using the high-order Nyström methods introduced in [1] for a bisinusoidal grating whose height to period ratio equals to 1.  ... 
doi:10.4171/owr/2010/10 fatcat:kckquuef3nandbkembon5szvr4

Nonstandard Finite Element Methods

Susanne Brenner, Carsten Carstensen, Peter Monk
2008 Oberwolfach Reports  
The mathematical analysis of standard conforming finite elements is very well advanced giving rise to highly efficient codes particularly for elliptic problems.  ...  These "non-standard" finite element methods are the subject of this Oberwolfach workshop. The extended abstracts here represent a snapshot of a varied and quickly evolving field.  ...  We consider a further, more detailed analysis of the discretization method as well as the iterative solver is a challenging task.  ... 
doi:10.4171/owr/2008/36 fatcat:n524fqo2ojafngsmk3xef2vtlm

Very singular similarity solutions and Hermitian spectral theory for semilinear odd-order PDEs

R. S. Fernandes, V. A. Galaktionov
2011 Journal of Partial Differential Equations  
Asymptotic large-and short-time behavior of solutions of the linear dispersion equation u t = u xxx in R×R + , and its (2k+1)th-order extensions are studied.  ...  Such a refined scattering is based on a "Hermitian" spectral theory for a pair {B,B * } of non self-adjoint rescaled operators l≥0}, where Ai(y) is Airy's classic function.  ...  and non-blow-up solutions and changing of the stability of the trivial zero solutions for the PDEs under consideration.  ... 
doi:10.4208/jpde.v24.n3.2 fatcat:ovrjjyy7w5e23nmoilrrwt46oi

Numerical homogenization beyond scale separation

Robert Altmann, Patrick Henning, Daniel Peterseim
2021 Acta Numerica  
Numerical homogenization is a methodology for the computational solution of multiscale partial differential equations.  ...  with a continuum of scales.  ...  We would like to thank Axel Målqvist and Houman Owhadi for their important input regarding historical remarks and the latest developments in the field.  ... 
doi:10.1017/s0962492921000015 fatcat:kxp53t5amvbl7d65fjfduvoyky

Exponential Integrators with Parallel-in-Time Rational Approximations for the Shallow-Water Equations on the Rotating Sphere [article]

Martin Schreiber, Nathanaël Schaeffer, Richard Loft
2019 arXiv   pre-print
This method replaces a time integration of stiff linear oscillatory and diffusive systems by the sum of the solutions of many decoupled systems, which can be solved in parallel.  ...  In this work we study a massively parallel rational approximation of exponential integrators (REXI).  ...  Acknowledgements Martin Schreiber received funding from NCAR for a research stay in summer 2017 at the Mesa Labs.  ... 
arXiv:1805.06557v3 fatcat:bcjwzmbrcrc4na6oza5nyqqi6y

COSMOLOGICAL EVOLUTION IN A TYPE-0 STRING THEORY

G. A. DIAMANDIS, B. C. GEORGALAS, N. E. MAVROMATOS, E. PAPANTONOPOULOS, I. PAPPA
2002 International Journal of Modern Physics A  
The Universe asymptotes, for large times, to a non-accelerating linearly-expanding Universe with a time-dependent dilaton and a relaxing to zero vacuum energy a la quintessence.  ...  We study the cosmological evolution of a type-0 string theory by employing non-criticality, which may be induced by fluctuations of the D3 brane worlds.  ...  Acknowledgements The work of E.P and I.P is partially supported by the NTUA program Archimedes.  ... 
doi:10.1142/s0217751x02010534 fatcat:3qhl4z7dnnas7hd57mkugwlok4

Wavenumber-explicit convergence analysis for finite element discretizations of time-harmonic wave propagation problems with perfectly matched layers

Théophile Chaumont-Frelet, Dietmar Gallistl, Serge Nicaise, Jérôme Tomezyk
2022 Communications in Mathematical Sciences  
The first part of this paper is devoted to a wavenumber-explicit stability analysis of a planar Helmholtz problem with a perfectly matched layer.  ...  We prove that, for a model scattering problem, the H 1 norm of the solution is bounded by the right-hand side, uniformly in the wavenumber k in the high wavenumber regime.  ...  We are interested in the highly oscillatory regime, which corresponds to large values of k. It is known that the finite element method (FEM) is not robust with respect to that parameter [1] .  ... 
doi:10.4310/cms.2022.v20.n1.a1 fatcat:5uayg55z55cnnh6hgxatoptokq
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