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String gradient weighted moving finite elements. (English) Zbl 1066.65104

Summary: Moving finite element methods are well established for the solution of systems of partial differential equations which contain regions where the solution is rapidly varying but moving. The string or second gradient weighted moving finite element method (SGWMFE) uses a piecewise linear discretization of a single evolving manifold to approximate the solution of the partial differential equations.
In the case of one space dimension, \(x\), and two dependent variables, \(u(x,t)\) and \(v(x,t)\), the solution is calculated from the normal motion of a single manifold \([x(\tau,t),u(\tau,t),v(\tau,t)]\), where \(\tau\) is a parameter along the maniflold, or a string embedded in \([x,u,v]\) space. This method can be extended to multiple dimensions and an arbitrary number of dependent variables in which case the string parameterization analogy is replaced by a multi-variable parameterization.
In this paper, we outline the application of SGWMFE for the solution of the shallow water equations in one and two space dimensions. We describe the results of a number of numerical experiments, including varying the initial distribution of nodes.

MSC:

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35K55 Nonlinear parabolic equations
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76M10 Finite element methods applied to problems in fluid mechanics
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References:

[1] Carlson, SIAM Journal on Scientific Computing 19 pp 728– (1998)
[2] Carlson, SIAM Journal on Scientific Computing 19 pp 766– (1998)
[3] Miller, SIAM Journal on Numerical Analysis 34 pp 67– (1997)
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[7] Miller, Journal of Computational and Applied Mathematics
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