Quasi optimal finite difference method for Helmholtz problem on unstructured grids

Fernandes, Daniel T.; Loula, Abimael F. D.

doi:10.1002/nme.2795

Cited by 12 publications

(17 citation statements)

References 39 publications

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“…Approaches based on polynomial basis coupled with nonstandard variational formulations (such as [75]) have been proposed in order to approximate the Helmholtz operator so that the resulting discrete problems have better stability properties. For example, with an appropriate choice of coefficients, low-order compact finite-difference discretizations can effectively reduce the dispersion error [35,58,80]. Other instances of such approaches are the generalized finite element method (GFEM) [4] and continuous interior penalty finite element method (CIP-FEM) [107,111], the interpolated optimized finite-difference method (IOFD) [93,94], Galerkin methods with hp refinement [70,72,73], among many others.…”

Section: Related Workmentioning

confidence: 99%

Learning dominant wave directions for plane wave methods for high-frequency Helmholtz equations

et al. 2017

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We present a ray-based finite element method for the high-frequency Helmholtz equation in smooth media, whose basis is learned adaptively from the medium and source. The method requires a fixed number of grid points per wavelength to represent the wave field; moreover, it achieves an asymptotic convergence rate of O(ωwhere ω is the frequency parameter in the Helmholtz equation. The local basis is motivated by the geometric optics ansatz and is composed of polynomials modulated by plane waves propagating in a few dominant ray directions. The ray directions are learned by processing a low-frequency wave field that probes the medium with the same source. Once the local ray directions are extracted, they are incorporated into the local basis to solve the high-frequency Helmholtz equation. This process can be continued to further improve the approximations for both local ray directions and high-frequency wave fields iteratively. Finally, a fast solver is developed for solving the resulting linear system with an empirical complexity O(ω d ) up to a poly-logarithmic factor. Numerical examples in 2D are presented to corroborate the claims.

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Section: Related Workmentioning

confidence: 99%

Learning dominant wave directions for plane wave methods for high-frequency Helmholtz equations

et al. 2017

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“…Using high precision float point operations, the expected accuracy and optimal convergence rates are recovered as observed in [28] and [30].…”

Section: Solving the Local Problemsmentioning

confidence: 98%

A quasi optimal Petrov–Galerkin method for Helmholtz problem

Loula

Fernandes

2009

Numerical Meth Engineering

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SUMMARYA Petrov-Galerkin finite element formulation is introduced for Helmholtz problem in two dimensions using polynomial weighting functions. At each node of the mesh, a global basis function for the weighting space is obtained adding to the bilinear C 0 Lagrangian weighting function linear combinations of polynomial bubbles defined on a macroelement containing this node. Quasi optimal weighting functions, with the same support of the corresponding global test functions, are obtained after computing the coefficients of these linear combinations attending to optimality criteria. This is done numerically through a simple preprocessing technique naturally applied to non-uniform and unstructured meshes. In particular, for uniform mesh a quasi optimal interior stencil of the same order of the quasi-stabilized finite element method stencil derived by Babuška et al. (Comput. Methods Appl. Mech. Engrg 1995; 128:325-359) is obtained. Numerical results are presented illustrating the great stability and accuracy of this formulation with uniform and non-uniform meshes.

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“…For convenience, Sutmann's scheme is named FD3D-6-27, in which the first number denotes the order of accuracy and the second denotes the total number of the involved points within a fundamental cube. Similarly, Fernandes and Loula also proposed a sixth-order accurate, quasi optimal finite difference method for Helmholtz problem [14] in three dimensions. Their coefficients are obtained numerically by minimizing a least-squares functional of the local truncation error for plane wave solutions in any direction.…”

Section: Introductionmentioning

confidence: 99%

Semi-Analytical Solutions of the 3-D Homogeneous Helmholtz Equation by the Method of Connected Local Fields

Chang

Mu²

2013

PIER

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Abstract-We advance the theory of the two-dimensional method of connected local fields (CLF) to the three-dimensional cases. CLF is suitable for obtaining semi-analytical solutions of Helmholtz equation. The fundamental building block (cell) of the 3-D CLF is a cube consisting of a central point and twenty six points on the cube's surface. These surface points form three symmetry groups: six on the planar faces, twelve on the edges and eight on the vertices (corners). The local field within the unit cell is expanded in a truncated spherical FourierBessel series. From this representation we develop a closed-form, 3D local field expansion (LFE) coefficients that relate the central point to its immediate neighbors. We also compute the CLF-based FD-FD numerical solutions of the 3D Green's function in free space. Compared with the analytic solution, we found that even at a low three points per wavelength spatial sampling, the accumulated phase errors of the CLF 3D Green's function after propagating a distance of ten wavelengths are well under ten percent.

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Quasi optimal finite difference method for Helmholtz problem on unstructured grids

Cited by 12 publications

References 39 publications

Learning dominant wave directions for plane wave methods for high-frequency Helmholtz equations

Learning dominant wave directions for plane wave methods for high-frequency Helmholtz equations

A quasi optimal Petrov–Galerkin method for Helmholtz problem

Semi-Analytical Solutions of the 3-D Homogeneous Helmholtz Equation by the Method of Connected Local Fields

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