Spectral element method

Summary

In the numerical solution of partial differential equations, a topic in mathematics, the spectral element method (SEM) is a formulation of the finite element method (FEM) that uses high degree piecewise polynomials as basis functions. The spectral element method was introduced in a 1984 paper by A. T. Patera. Although Patera is credited with development of the method, his work was a rediscovery of an existing method (see Development History) Discussion The spectral method expands the solution in trigonometric series, a chief advantage being that the resulting method is of a very high order. This approach relies on the fact that trigonometric polynomials are an orthonormal basis for L^2(\Omega). The spectral element method chooses instead a high degree piecewise polynomial basis functions, also achieving a very high order of accuracy. Such polynomials are usually orthogonal Chebyshev polynomials or very high order Lagrange polynomials over non-uniformly spac

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Stabilized spectral element methods for the simulation of viscoelastic flows

EPFL2002

High order discontinuous Galerkin methods on simplicial elements for the elastodynamics equation

Carlo Marcati, Alfio Quarteroni

In this work we apply the discontinuous Galekin (dG) spectral element method on meshes made of simplicial elements for the approximation of the elastodynamics equation. Our approach combines the high accuracy of spectral methods, the geometrical flexibility of simplicial elements and the computational efficiency of dG methods. We analyze the dissipation, dispersion and stability properties of the resulting scheme, with a focus on the choice of different sets of basis functions. Finally, we apply the method on benchmark as well as realistic test cases.

Springer Verlag2016

Simulation of flows of viscoelastic fluids at high Weissenberg number using a filter-based stabilization of the spectral element method

Michel Deville, Nicolas Fietier, Azadeh Jafari

The challenge for computational rheologists is to develop efficient and stable numerical schemes in order to obtain accurate numerical solutions for the governing equations at values of practical interest of the Weissenberg number. One of the associated problems for numerical simulation of viscoelastic fluids is that the accuracy of the results when approaching critical values at which numerical instabilities occur is very low and refining the mesh proved to be not very helpful. In order to investigate the numerical instability generation a comprehensive study about the growth of spurious modes with time evolution, mesh refinement, boundary conditions and Weissenberg number or any other affected parameters is performed on the planar Poiseuille channel flow. To get rid of these spurious modes the filter based stabilization of spectral element methods proposed by Boyd (1998) [1] is applied. This filter technique is very useful to eliminate spurious modes for one element decomposition, while in the case of multi-element configuration, the performance of this technique is not ideal. Since the performance of filter-based stabilization of spectral element acts very well for one element decomposition, a possible remedy to solve the associated problem of multi-element decomposition is mesh-transfer technique which means: first mapping the multi-element configuration to one element configuration, applying filter-based stabilization technique to this new topology and hereafter transferring the filtered variables to the original configuration. This way of implementing filtering is very useful for the Oldroyd-B fluids when a moderate number of grid points is used. (C) 2011 Elsevier Ltd. All rights reserved.

2012