The h, p and h-p version of the finite element method; basis theory and applications

Babuška, Ivo; Guo, Benqi

doi:10.1016/0965-9978(92)90097-y

Cited by 116 publications

(99 citation statements)

References 41 publications

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“…With this method the dispersion error is drastically reduced [8] and it has been shown to be a valid approach to address the pollution effect and tackle large-scale problems [9,10]. The p-FEM approach provides exponential convergence when increasing the polynomial order p. It is also well-suited for p-adaptivity where the polynomial order is adjusted locally in the computational domain [11]. Finally, the hierarchic nature of the Lobatto shape functions leads to efficient algorithms for solving the same model over a range of frequencies [10].…”

Section: Introductionmentioning

confidence: 99%

A comparison of high-order polynomial and wave-based methods for Helmholtz problems

Lieu

Gabard

Bériot

2016

Journal of Computational Physics

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The application of computational modelling to wave propagation problems is hindered by the dispersion error introduced by the discretisation. Two common strategies to address this issue are to use high-order polynomial shape functions (e.g. hp-FEM), or to use physics-based, or Trefftz, methods where the shape functions are local solutions of the problem (typically plane waves). Both strategies have been actively developed over the past decades and both have demonstrated their benefits compared to conventional finite-element methods, but they have yet to be compared. In this paper a high-order polynomial method (p-FEM with Lobatto polynomials) and the wave-based discontinuous Galerkin method are compared for two-dimensional Helmholtz problems. A number of different benchmark problems are used to perform a detailed and systematic assessment of the relative merits of these two methods in terms of interpolation properties, performance and conditioning. It is generally assumed that a wave-based method naturally provides better accuracy compared to polynomial methods since the plane waves or Bessel functions used in these methods are exact solutions of the Helmholtz equation. Results indicate that this expectation does not necessarily translate into a clear benefit, and that the differences in performance, accuracy and conditioning are more nuanced than generally assumed. The high-order polynomial method can in fact deliver comparable, and in some cases superior, performance compared to the wave-based DGM. In addition to benchmarking the intrinsic computational performance of these methods, a number of practical issues associated with realistic applications are also discussed.

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Section: Introductionmentioning

confidence: 99%

A comparison of high-order polynomial and wave-based methods for Helmholtz problems

Lieu

Gabard

Bériot

2016

Journal of Computational Physics

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“…This graph also shows that (1) for each discretization, the estimator given by Eq. (4) overestimates the error; (2) as expected, the magnitude of the errors for the finer mesh (72 nodes) are smaller than the corresponding ones for the coarser mesh (36 nodes); ( 3 ) the difference between the estimated and exact errors are smaller for the finer mesh (72 nodes) than for the coarser one (36 nodes); and (4) the symmetry of the problem is captured by both the BEM solution (Eq. (1)) and the hypersingular BIE for the boundary error estimate (Eq.…”

Section: Circlementioning

confidence: 83%

A posteriori pointwise error estimates for the boundary element method

Paulino

Gray

Zarikian

1995

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“…Such a solution can be compared to typical 2D solutions near re-entrant corners: u = r β φ(θ) for α = β + 1/2. The problem is solved using h and p finite element approximations, with both homogeneous or geometric meshes [27]. The finite element shape functions are based on integrated Legendre polynomials, as presented in [1].…”

Section: D Model Problemmentioning

confidence: 99%

Treatment of nearly-singular problems with the X-FEM

Legrain

Moës

2014

Adv. Model. and Simul. in Eng. Sci.

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Background: In recent years, lot of research have been conducted on fictitious domain approaches in order to simplify the meshing process for computed aided analysis. The behaviour of such non-conforming methods is studied in the case of the approximation of nearly singular solutions. Such solutions appear when problems involve singularities whose center are located outside (but close) of the domain of interest. These solutions are common in industrial structures that usually involve rounded re-entrant corners. Methods: The performance of the finite element method is evaluated in this context by means of a simple unidimensional example. Both numerical and theoretical estimates are considered in order to assess the behaviour of the numerical approximation. It is demonstrated that despite being regular, the convergence of the approximation can be bounded to an algebraic rate that depends on the solution. Reasons for such behaviour are presented, and two complementary strategies are proposed in order to recover optimal convergence rates. The first strategy is based on a proper enrichment of the approximation thanks to the X-FEM, while the second is based on a proper mesh design that follows a geometric progression. Finally, the proposed strategies are extended and validated in 2D. Results: The performance of the two strategies is highlighted for both 1D and 2D examples. Both methods allow to recover proper convergence rates (optimal algebraic rate for h-convergence, exponential for p-convergence) in 1D and 2D. Conclusions: The proposed strategies allow for a very accurate solution for such solutions. The enrichment strategy is valid for both h and p refinement, whereas the mesh-design strategy is only usable for p refinement. However, such enrichment functions can be tedious to derive.

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The h, p and h-p version of the finite element method; basis theory and applications

Cited by 116 publications

References 41 publications

A comparison of high-order polynomial and wave-based methods for Helmholtz problems

A comparison of high-order polynomial and wave-based methods for Helmholtz problems

A posteriori pointwise error estimates for the boundary element method

Treatment of nearly-singular problems with the X-FEM

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