A least-squares finite element method for the Helmholtz equation

Chang, Ching L.

doi:10.1016/0045-7825(90)90121-2

Cited by 42 publications

(29 citation statements)

References 7 publications

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“…As we shall see below, for (2.12), optimal convergence in the H 1 (Ω)-norm requires an additional "grid decomposition property" on the finite element spaces (see [61] and [72]) or an additional "curl" constraint added to the first-order system (2.11) (see [43], [44], [48], [49], [95], and [106].) Other examples of first-order systems that are not H 1 -coercive are given by particular decompositions of the biharmonic equation and the Stokes problem.…”

Section: A Recipe For Practicalitymentioning

confidence: 99%

“…In view of the equation v = grad φ, one can augment (2.11) by a compatibility condition known as the curl constraint (see, e.g., [55], [49], [48], and [95])…”

Section: Discretization Levelmentioning

confidence: 99%

“…As a result, least-squares finite element methods in these settings are among the best understood, studied, and tested from both the theoretical and computational viewpoints. Our discussion will also include least-squares methods for convection-diffusion and other second-order elliptic problems (see [6], [24], [26], [33], [35], [38], [42]- [45], [48], [52], [62], [71], [72], [75], [86], [95], [106], [107], and [104]), linear elasticity (see [34], [36], and [37]), inviscid, compressible flows (see [61], [64], [67], [99], and [118]), and electromagnetics (see [46], [58], [60], [97], and [116]. )…”

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confidence: 99%

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Finite Element Methods of Least-Squares Type

Bochev¹,

Gunzburger²

1998

SIAM Rev.

285

237

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Abstract. We consider the application of least-squares variational principles to the numerical solution of partial differential equations. Our main focus is on the development of least-squares finite element methods for elliptic boundary value problems arising in fields such as fluid flows, linear elasticity, and convection-diffusion. For many of these problems, least-squares principles offer numerous theoretical and computational advantages in the algorithmic design and implementation of corresponding finite element methods, that are not present in standard Galerkin discretizations. Most notably, the use of least-squares principles leads to symmetric and positive definite algebraic problems and allows one to circumvent stability conditions such as the inf-sup condition arising in mixed methods for the Stokes and Navier-Stokes equations. As a result, application of least-squares principles has led to the development of robust and efficient finite element methods for a large class of problems of practical importance.

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Section: A Recipe For Practicalitymentioning

confidence: 99%

“…In view of the equation v = grad φ, one can augment (2.11) by a compatibility condition known as the curl constraint (see, e.g., [55], [49], [48], and [95])…”

Section: Discretization Levelmentioning

confidence: 99%

mentioning

confidence: 99%

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Finite Element Methods of Least-Squares Type

Bochev¹,

Gunzburger²

1998

SIAM Rev.

285

237

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show abstract

“…The idea of adding the curl constraint in developing a least-squares functional for Poisson's equation has been used by several researchers (see, e.g., [13,16,17,23,14,15,24,30]). For Poisson's equation, the key tool is the proof that (H(div) ∩ H(curl)) is algebraically and topologically imbedded in (H 1 ) n , which was developed by Girault and Raviart [20] for problems with strictly Dirichlet or Neumann boundary conditions.…”

mentioning

confidence: 99%

First-Order System Least Squares for Second-Order Partial Differential Equations: Part II

Cai¹,

Manteuffel²,

McCormick³

1997

SIAM J. Numer. Anal.

180

212

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Abstract. This paper develops a least-squares functional that arises from recasting general second-order uniformly elliptic partial differential equations in n = 2 or 3 dimensions as a system of first-order equations. In part I [Z. Cai, R. D. Lazarov, T. Manteuffel, and S. McCormick, SIAM J. Numer. Anal., 31 (1994), pp. 1785-1799] a similar functional was developed and shown to be elliptic in the H(div) × H 1 norm and to yield optimal convergence for finite element subspaces of H(div)×H 1 . In this paper the functional is modified by adding a compatible constraint and imposing additional boundary conditions on the first-order system. The resulting functional is proved to be elliptic in the (H 1 ) n+1 norm. This immediately implies optimal error estimates for finite element approximation by standard subspaces of (H 1 ) n+1 . Another direct consequence of this ellipticity is that multiplicative and additive multigrid algorithms applied to the resulting discrete functionals are optimally convergent. As an alternative to perturbation-based approaches, the least-squares approach developed here applies directly to convection-diffusion-reaction equations in a unified way and also admits a fast multigrid solver, historically a missing ingredient in least-squares methodology.Key words. least-squares discretization, multigrid, second-order elliptic problems, iterative methods AMS subject classifications. 65F10, 65F30PII. S00361429942660661. Introduction. The object of study of this paper, and its earlier companion [11], is the solution of elliptic equations (including convection-diffusion and Helmholtz equations) by way of a least-squares formulation for an equivalent first-order system. Such formulations have been considered by several researchers over the last few decades (see the historical discussion in [11]), motivated in part by the possibility of a well-posed variational principle for a general class of problems. In [11] a similar functional was developed and shown to be elliptic in the H(div) × H 1 norm and to yield optimal convergence for finite element subspaces of H(div) × H 1 . In this paper the functional is modified by adding a compatible constraint and imposing additional boundary conditions on the first-order system. It is shown that the resulting functional is elliptic in the (H 1 ) n+1 norm. Direct consequences of this result are optimal approximation error estimates for standard finite element subspaces of (H 1 ) n+1 and optimal convergence of multiplicative and additive multigrid algorithms applied to the resulting discrete functionals. As an alternative to perturbation-based approaches (cf. [1,3,9,10,25,34,35]), the least-squares approach developed here applies directly to convection-diffusion-reaction equations in a unified way and also admits an efficient multilevel solver, historically a missing ingredient in least-squares methodology.

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“…Finite element methods for acoustic wave propagation problems such as (1.1a)-(1.1c) have been widely studied in the literature (cf., e.g., [4,12,15,28,30] as well as the survey article [17], the monographs [27,29] and the references therein). In case of large wavenumbers k, the finite element discretization typically requires fine meshes for a proper resolution of the waves and thus results in large linear algebraic systems to be solved.…”

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confidence: 99%

Convergence analysis of an adaptive interior penalty discontinuous Galerkin method for the Helmholtz equation

Hoppe

Sharma

2012

IMA Journal of Numerical Analysis

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Abstract. We are concerned with a convergence analysis of an adaptive Interior Penalty Discontinuous Galerkin (IPDG) method for the numerical solution of acoustic wave propagation problems as described by the Helmholtz equation. The mesh adaptivity relies on a residual-type a posteriori error estimator that does not only control the approximation error but also the consistency error caused by the nonconformity of the approach. As in the case of IPDG for standard second order elliptic boundary value problems, the convergence analysis is based on the reliability of the estimator, an estimator reduction property, and a quasi-orthogonality result. However, in contrast to the standard case, special attention has to be paid to a proper treatment of the lower order term in the equation containing the wavenumber which is taken care of by an Aubin-Nitsche type argument for the associated conforming finite element approximation. Numerical results are given for an interior Dirichlet problem and a screen problem illustrating the performance of the adaptive IPDG method.

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A least-squares finite element method for the Helmholtz equation

Cited by 42 publications

References 7 publications

Finite Element Methods of Least-Squares Type

Finite Element Methods of Least-Squares Type

First-Order System Least Squares for Second-Order Partial Differential Equations: Part II

Convergence analysis of an adaptive interior penalty discontinuous Galerkin method for the Helmholtz equation

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