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47
Numerical solution of the Helmholtz equation with high wave numbers
 International Journal for Numerical Methods in Engineering 2004
"... This paper investigates the pollution effect, and explores the feasibility of a local spectral method, the discrete singular convolution (DSC) algorithm for solving the Helmholtz equation with high wavenumbers. Fourier analysis is employed to study the dispersive error of the DSC algorithm. Our anal ..."
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Cited by 31 (8 self)
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This paper investigates the pollution effect, and explores the feasibility of a local spectral method, the discrete singular convolution (DSC) algorithm for solving the Helmholtz equation with high wavenumbers. Fourier analysis is employed to study the dispersive error of the DSC algorithm. Our analysis of dispersive errors indicates that the DSC algorithm yields a dispersion vanishing scheme. The dispersion analysis is further confirmed by the numerical results. For one and higherdimensional Helmholtz equations, the DSC algorithm is shown to be an essentially pollutionfree scheme. Furthermore, for largescale computation, the grid density of the DSC algorithm can be close to the optimal two grid points per wavelength. The present study reveals that the DSC algorithm is accurate and efficient for solving the Helmholtz equation with high wavenumbers. Copyright � 2003 John Wiley & Sons, Ltd. KEY WORDS: Helmholtz equation; high wavenumber; pollution effect; discrete singular convolution; dispersion analysis 1.
Metric based upscaling
 Communications on Pure and Applied Mathematics
, 2007
"... We consider divergence form elliptic operators in dimension n ≥ 2. Although solutions of these operators are only Hölder continuous, we show that they are differentiable (C 1,α) with respect to harmonic coordinates. It follows that numerical homogenization can be extended to situations where the med ..."
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Cited by 23 (2 self)
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We consider divergence form elliptic operators in dimension n ≥ 2. Although solutions of these operators are only Hölder continuous, we show that they are differentiable (C 1,α) with respect to harmonic coordinates. It follows that numerical homogenization can be extended to situations where the medium has no ergodicity at small scales and is characterized by a continuum of scales by transferring a new metric in addition to traditional averaged (homogenized) quantities from subgrid scales into computational scales and error bounds can be given. This numerical homogenization method can also be used as a compression tool for differential operators. 1 Introduction and main results Let Ω be a bounded and convex domain of class C2. We consider the following benchmark PDE
Multiscale and stabilized methods
 ENCYCLOPEDIA OF COMPUTATIONAL MECHANICS
, 2004
"... This article presents an introduction to multiscale and stabilized methods, which represent unied approaches to modeling and numerical solution of fluid dynamic phenomena. Finite element applications are emphasized but the ideas are general and apply to other numerical methods as well. (They have be ..."
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Cited by 22 (8 self)
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This article presents an introduction to multiscale and stabilized methods, which represent unied approaches to modeling and numerical solution of fluid dynamic phenomena. Finite element applications are emphasized but the ideas are general and apply to other numerical methods as well. (They have been used in the development of finite difference, nite volume, and spectral methods, in addition to nite element methods.) The analytical ideas are rst illustrated for timeharmonic wavepropagation problems in unbounded fluid domains governed by the Helmholtz equation. This leads to the wellknown DirichlettoNeumann formulation. A general treatment of the variational multiscale method in the context of an abstract Dirichlet problem is then presented which is applicable to advectivediffusive processes and other processes of physical interest. It is shown how the exact theory represents a paradigm for subgridscale models and a posteriori error estimation. Hierarchical pmethods and bubble function methods are examined in order to understand and, ultimately, approximate the "finescale Green's function " which appears in the theory. Relationships among socalled residualfree bubbles, element Green's functions, and stabilized methods are exhibited. These ideas are then generalized to a class of nonsymmetric, linear evolution operators formulated in space
Multiscale enrichment based on partition of unity
 Inter. J. Numer. Meth. Engrg
"... A new Multiscale Enrichment method based on the Partition of Unity (MEPU) method is presented. It is a synthesis of mathematical homogenization theory and the Partition of Unity method. Its primary objective is to extend the range of applicability of mathematical homogenization theory to problems wh ..."
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Cited by 21 (1 self)
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A new Multiscale Enrichment method based on the Partition of Unity (MEPU) method is presented. It is a synthesis of mathematical homogenization theory and the Partition of Unity method. Its primary objective is to extend the range of applicability of mathematical homogenization theory to problems where scale separation may not be possible. MEPU is perfectly suited for enriching the coarse scale continuum descriptions (PDEs) with fine scale features and the quasicontinuum formulations with relevant atomistic data. Numerical results show that it provides a considerable improvement over classical mathematical homogenization theory and quasicontinuum formulations. 1.
Fast integration and weight function blending in the extended finite element method
, 2009
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Blending in the extended finite element method by discontinuous Galerkin and assumed strain methods
, 2007
"... In the extended finite element method (XFEM), errors are caused by parasitic terms in the approximation space of the blending elements at the edge of the enriched subdomain. A discontinuous Galerkin (DG) formulation is developed, which circumvents this source of error. A patchbased version of the D ..."
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Cited by 15 (3 self)
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In the extended finite element method (XFEM), errors are caused by parasitic terms in the approximation space of the blending elements at the edge of the enriched subdomain. A discontinuous Galerkin (DG) formulation is developed, which circumvents this source of error. A patchbased version of the DG formulation is developed, which decomposes the domain into enriched and unenriched subdomains. Continuity between patches is enforced with an internal penalty method. An elementbased form is also developed, where each element is considered a patch. The patchbased DG is shown to have similar accuracy to the elementbased DG for a given discretization but requires significantly fewer degrees of freedom. The method is applied to material interfaces, cracks and dislocation problems. For the dislocations, a contour integral form of the internal forces that only requires integration over the patch boundaries is developed. A previously developed assumed strain (AS) method is also developed further and compared with the DG method for weak discontinuities and linear elastic cracks. The DG method is shown to be significantly more accurate than the standard XFEM for a given element size and to converge
Preasymptotic error analysis of CIPFEM and FEM for the Helmholtz equation with high wave number. Part I: linear version
, 2014
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Midfrequency vibroacoustic modelling: challenges and potential solutions
 In Proceedings of ISMA 2002
, 2002
"... At present, the main numerical modelling techniques for acoustic and (coupled) vibroacoustic analysis are based on element based techniques, such as the finite element and boundary element method. Due to the huge computational efforts, the use of these deterministic techniques is practically restri ..."
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Cited by 10 (5 self)
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At present, the main numerical modelling techniques for acoustic and (coupled) vibroacoustic analysis are based on element based techniques, such as the finite element and boundary element method. Due to the huge computational efforts, the use of these deterministic techniques is practically restricted to lowfrequency applications. For highfrequency modelling, some alternative, probabilistic techniques such as SEA have been developed. However, there is still a wide midfrequency range, for which no adequate and mature prediction techniques are available at the moment. In this frequency range, the computational efforts of conventional element based techniques become prohibitively large, while the basic assumptions of the probabilistic techniques are not yet valid. In recent years, a vast amount of research has been initiated in a quest for an adequate solution for the current midfrequency problem. This paper discusses the various methodologies that are being explored in this perspective. The main focus of this paper lies on the methodology that looks for deterministic techniques with an enhanced convergence rate and computational efficiency compared to the conventional element based methods in order to shift the practical frequency limitation towards the midfrequency range. In this respect, special attention is paid to the wave based prediction technique for (coupled) vibroacoustic analysis that is being developed at the KULeuven Noise and Vibration Research group. The method is based on an indirect Trefftz approach. Various recent validations have revealed the beneficial convergence rate of this novel technique, thereby exhibiting its potential to comply with the midfrequency modelling challenge. 1.
Stabilized finite element methods based on multiscale enrichment for the Stokes problem
 SIAM J. Numer. Anal
, 2006
"... Abstract. This work concerns the development of stabilized finite element methods for the Stokes problem considering nonstable different (or equal) order of velocity and pressure interpolations. The approach is based on the enrichment of the standard polynomial space for the velocity component with ..."
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Abstract. This work concerns the development of stabilized finite element methods for the Stokes problem considering nonstable different (or equal) order of velocity and pressure interpolations. The approach is based on the enrichment of the standard polynomial space for the velocity component with multiscale functions which no longer vanish on the element boundary. On the other hand, since the test function space is enriched with bubblelike functions, a Petrov–Galerkin approach is employed. We use such a strategy to propose stable variational formulations for continuous piecewise linear in velocity and pressure and for piecewise linear/piecewise constant interpolation pairs. Optimal order convergence results are derived and numerical tests validate the proposed methods. Key words. Stokes equation, multiscale functions, SIMPLEST element, bubble function