On different finite element methods for approximating the gradient of the solution to the helmholtz equation

Haslinger, Jaroslav; Neittaanmäki, Pekka

doi:10.1016/0045-7825(84)90022-7

Cited by 14 publications

(9 citation statements)

References 13 publications

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“…Chang [10], Chen [11], Fix, Gunzburger and Nicolaides [15], Haslinger and Neittaanmäki [19], Neittaanmäki and Saranen [25], Jespersen [20]. The results in these papers are discussed in detail in Pehlivanov and Carey [28], see also Carey et al [6].…”

Section: Introductionmentioning

confidence: 74%

Least-squares mixed finite element methods for non-selfadjoint elliptic problems: I. Error estimates

Pehlivanov

Carey

Vassilevski

1996

Numerische Mathematik

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A least-squares mixed finite element method for general secondorder non-selfadjoint elliptic problems in two-and three-dimensional domains is formulated and analyzed. The finite element spaces for the primary solution approximation u h and the flux approximation σ h consist of piecewise polynomials of degree k and r respectively. The method is mildly nonconforming on the boundary. The cases k = r and k + 1 = r are studied. It is proved that the method is not subject to the LBB-condition. Optimal L 2 -and H 1 -error estimates are derived for regular finite element partitions. Numerical experiments, confirming the theoretical rates of convergence, are presented.

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Section: Introductionmentioning

confidence: 74%

Least-squares mixed finite element methods for non-selfadjoint elliptic problems: I. Error estimates

Pehlivanov

Carey

Vassilevski

1996

Numerische Mathematik

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“…Moreover, the L 2 -and H 1 -error estimates do not require the finite element mesh to be quasi-uniform. Related studies on least squares mixed methods for elliptic equations can be found in [4,6,11,12,13,16] and references therein.…”

Section: Introduction Consider the Initial Boundary Value Problemmentioning

confidence: 99%

AnH¹-Galerkin Mixed Finite Element Method for an Evolution Equation with a Positive-Type Memory Term

Pani¹,

Fairweather²

2002

SIAM J. Numer. Anal.

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An H 1 -Galerkin mixed finite element method is analyzed for a class of evolution equations with memory. When a classical mixed method is applied to such problems, it has not been possible to obtain any estimate for the flux. However, the proposed approach yields optimal order convergence without the LBB consistency condition and quasi uniformity of the finite element mesh. Compared to the results proved for one space variable, the L 2 estimate of the flux is not optimal for problems in two and three space dimensions. Therefore, a modification of the method is proposed and analyzed. A maximum norm estimate is also derived in one and two space variables. A backward Euler approximation of the modified method is also analyzed.

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“…Studies based on first-order system formulations of the Helmholtz equation include [16], [18], [21], and [23]. The approach described in [16] does not include a curl term in the functional, so its discretizations are somewhat restrictive and its standard multigrid methods could not perform well with large wavenumbers, as expected.…”

mentioning

confidence: 99%

“…The approach described in [16] does not include a curl term in the functional, so its discretizations are somewhat restrictive and its standard multigrid methods could not perform well with large wavenumbers, as expected. The functional in [18] incorporates a curl expression but not in a way that achieves uniformity in the discretization (or the multigrid solver, had they analyzed it). In fact, except for [21] and [23] which is the basis of the work presented in the present paper, none of the methods cited above were shown to achieve optimal discretization accuracy and multigrid convergence.…”

mentioning

confidence: 99%

First-Order System Least-Squares for the Helmholtz Equation

Lee¹,

Manteuffel²,

McCormick³

et al. 2000

SIAM J. Sci. Comput.

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This paper develops a multilevel least-squares approach for the numerical solution of the complex scalar exterior Helmholtz equation. This second-order equation is first recast into an equivalent first-order system by introducing several "field" variables. A combination of scaled L 2 and H −1 norms is then applied to the residual of this system to create a least-squares functional. It is shown that, in an appropriate Hilbert space, the homogeneous part of this functional is equivalent to a squared graph norm, that is, a product norm on the space of individual variables. This equivalence to a norm that decouples the variables means that standard finite element discretization techniques and standard multigrid solvers can be applied to obtain optimal performance. However, this equivalence is not uniform in the wavenumber k, which can signal degrading performance of the numerical solution process as k increases. To counter this difficulty, we obtain a result that characterizes the error components causing performance degradation. We do this by defining a finite-dimensional subspace of these components on whose orthogonal complement k-uniform equivalence is proved for this functional and an analogous functional that is based only on L 2 norms. This subspace equivalence motivates a nonstandard multigrid method that attempts to achieve optimal convergence uniformly in k. We report on numerical experiments that empirically confirm k-uniform optimal performance of this multigrid solver. We also report on tests of the error in our discretization that seem to confirm optimal accuracy that is free of the so-called pollution effect.

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On different finite element methods for approximating the gradient of the solution to the helmholtz equation

Cited by 14 publications

References 13 publications

Least-squares mixed finite element methods for non-selfadjoint elliptic problems: I. Error estimates

Least-squares mixed finite element methods for non-selfadjoint elliptic problems: I. Error estimates

AnH¹-Galerkin Mixed Finite Element Method for an Evolution Equation with a Positive-Type Memory Term

First-Order System Least-Squares for the Helmholtz Equation

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On different finite element methods for approximating the gradient of the solution to the helmholtz equation

Cited by 14 publications

References 13 publications

Least-squares mixed finite element methods for non-selfadjoint elliptic problems: I. Error estimates

Least-squares mixed finite element methods for non-selfadjoint elliptic problems: I. Error estimates

AnH1-Galerkin Mixed Finite Element Method for an Evolution Equation with a Positive-Type Memory Term

First-Order System Least-Squares for the Helmholtz Equation

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Product

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AnH¹-Galerkin Mixed Finite Element Method for an Evolution Equation with a Positive-Type Memory Term