Computing with hp-ADAPTIVE FINITE ELEMENTS

Demkowicz, Leszek; Kurtz, Jason; Pardo, David; Paszenski, Maciek; Rachowicz, Waldemar; Zdunek, Adam

doi:10.1201/9781420011692

Cited by 212 publications

(85 citation statements)

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“…A construction of the Galerkin basis (6) builds up on a set of Nédélec elements of the uniform order p. A detailed description is found in [8]. Let the surface be given by a regular mesh consisting of curvilinear triangular boundary elements D…”

Section: Galerkin Bemmentioning

confidence: 99%

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Matrix valued adaptive cross approximation

Rjasanow

Weggler

2016

Math Methods in App Sciences

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A new variant of the Adaptive Cross Approximation (ACA) for approximation of dense block matrices is presented. This algorithm can be applied to matrices arising from the Boundary Element Methods (BEM) for elliptic or Maxwell systems of partial differential equations. The usual interpolation property of the ACA is generalised for the matrix valued case. Some numerical examples demonstrate the efficiency of the new method. The main example will be the electromagnetic scattering problem, that is, the exterior boundary value problem for the Maxwell system. Here, we will show that the matrix valued ACA method works well for high order BEM, and the corresponding high rate of convergence is preserved. Another example shows the efficiency of the new method in comparison with the standard technique, whilst approximating the smoothed version of the matrix valued fundamental solution of the time harmonic Maxwell system. Copyright © 2016 John Wiley & Sons, Ltd.

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Section: Galerkin Bemmentioning

confidence: 99%

“…On the reference triangle O T, the set of Nédélec elements of the first kind [8,9] is defined, that is,…”

Section: Galerkin Bemmentioning

confidence: 99%

Matrix valued adaptive cross approximation

Rjasanow

Weggler

2016

Math Methods in App Sciences

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“…The higher-order inset, however, allows to evaluate values for the fields and the field gradients at an order of convergence that is equal to the convergence order of the overall FE solution for the magnetic vector potential. This approach can be organised in an hp-adaptive FE package [17]. More elaborated approaches where the region of interest is modelled by an analytic solution, feature optimal accuracy [14] but require specialised algebraic solution techniques to preserve an acceptable computational efficiency [18].…”

Section: Hybrid Resolutionmentioning

confidence: 99%

Improved field post‐processing for a Stern–Gerlach magnetic deflection magnet

Gersem

Masschaele

Roggen

et al. 2013

Int J Numerical Modelling

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SUMMARYA Stern–Gerlach magnet applies a field gradient in order to deflect particles such that their magnetic moment can be determined. The design of such magnets is based on finite‐element (FE) simulations of the magnetic field and its gradient. However, inaccuracies arise when these quantities are calculated from an FE solution for the magnetic vector potential by numerical differentiation. For first‐order FE shape functions, the discretisation error for the field gradient may even fail to converge. An improved post‐processing approach marks a region in the magnet aperture where the field is post‐processed by matching a local analytical solution with the obtained FE solution. The local solution allows to derive values for the magnetic field and its gradient without reducing the convergence order of the discretisation error. The method is validated against the analytic solution for a two‐wire configuration and is applied for the design of a Rabi‐type Stern–Gerlach magnet. Copyright © 2013 John Wiley & Sons, Ltd.

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“…In the direct-current (DC) resistivity numerical modelling, the finite-element methods (FEMs) (e.g., Hughes (1987); Zienkiewicz and Taylor (2000); Wu (2003); Rücker, Günther, and Spitzer (2006); Cardarelli and Fischanger (2006); Demkowicz (2007); and Qiang, Han, and Dai (2013)) with a capability of dealing with complex geological models are now recognized as the most efficient numerical tool, as compared with other numerical methods such as the integral equation methods (e.g., Lee (1975) and Hvozdara and Kaikkonen (1996)), the boundary element methods (Xu, Zhao, and Yi 1998), and the finite-difference methods (e.g., Mufti (1976); Dey and Morrison (1979); Spitzer (1995); Mundry (1984); * E-mail: qiangjianke@163.com and Moucha and Bailey (2004)). In the FEM approach, the computation time heavily depends on the number of grid points.…”

Section: Introductionmentioning

confidence: 99%

2.5D direct‐current resistivity forward modelling and inversion by finite‐element–infinite‐element coupled method

Yuan

Tang

et al. 2015

Geophysical Prospecting

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To reduce the numerical errors arising from the improper enforcement of the artificial boundary conditions on the distant surface that encloses the underground part of the subsurface, we present a finite‐element–infinite‐element coupled method to significantly reduce the computation time and memory cost in the 2.5D direct‐current resistivity inversion. We first present the boundary value problem of the secondary potential. Then, a new type of infinite element is analysed and applied to replace the conventionally used mixed boundary condition on the distant boundary. In the internal domain, a standard finite‐element method is used to derive the final system of linear equations. With a novel shape function for infinite elements at the subsurface boundary, the final system matrix is sparse, symmetric, and independent of source electrodes. Through lower upper decomposition, the multi‐pole potentials can be swiftly obtained by simple back‐substitutions. We embed the newly developed forward solution to the inversion procedure. To compute the sensitivity matrix, we adopt the efficient adjoint equation approach to further reduce the computation cost. Finally, several synthetic examples are tested to show the efficiency of inversion.

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Computing with hp-ADAPTIVE FINITE ELEMENTS

Cited by 212 publications

References 0 publications

Matrix valued adaptive cross approximation

Matrix valued adaptive cross approximation

Improved field post‐processing for a Stern–Gerlach magnetic deflection magnet

2.5D direct‐current resistivity forward modelling and inversion by finite‐element–infinite‐element coupled method

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