The Lagrange-Galerkin spectral element method on unstructured quadrilateral grids
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The purpose of this paper is to introduce a new method formed by fusing the Lagrange-Galerkin and spectral element methods. The Lagrange-Galerkin method traces the characteristic curves of the solution and, consequently, is very well suited for resolving the nonlinearities introduced by the advection operator of the fluid dynamics equations. Spectral element methods are essentially higher order finite element methods that exhibit spectral (exponential) convergence, provided that the solution is a smooth function. By combining these two methods, a numerical scheme can be constructed that resolves, with extremely high precision, the nonlinearities of the advection terms and the smooth regions of the flow generated by the diffusion terms. This paper describes the construction of the Lagrange-Galerkin spectral element method which permits the use of any grid type including unstructured grids. The only restriction at the moment is that the grid elements be quadrilaterals. The stability analysis of both methods demonstrates why these two methods are so powerful individually and how their fusion leads to an improved scheme. The Lagrange-Galerkin spectral element method is validated using the 1D and 2D advection and advection-diffusion equations. The results of the stability analysis and the numerical experiments demonstrate the utility of such an approach. (C) 1998 Academic Press.
The article of record as published may be located at http://dx.doi.org/10.1006/jcph.1998.6078
RightsThis publication is a work of the U.S. Government as defined in Title 17, United States Code, Section 101. Copyright protection is not available for this work in the United States.
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