Math @ Duke

Publications [#268314] of John E. Dolbow
Papers Published
 Mourad, HM; Dolbow, J; Garikipati, K, An assumedgradient finite element method for the level set equation,
International Journal for Numerical Methods in Engineering, vol. 64 no. 8
(2005),
pp. 10091032, ISSN 00295981 [1395], [doi]
(last updated on 2018/07/18)
Abstract: The level set equation is a nonlinear advection equation, and standard finiteelement and finitedifference strategies typically employ spatial stabilization techniques to suppress spurious oscillations in the numerical solution. We recast the level set equation in a simpler form by assuming that the level set function remains a signed distance to the front/interface being captured. As with the original level set equation, the use of an extensional velocity helps maintain this signeddistance function. For some interfaceevolution problems, this approach reduces the original level set equation to an ordinary differential equation that is almost trivial to solve. Further, we find that sufficient accuracy is available through a standard Galerkin formulation without any stabilization or discontinuitycapturing terms. Several numerical experiments are conducted to assess the ability of the proposed assumedgradient level set method to capture the correct solution, particularly in the presence of discontinuities in the extensional velocity or levelset gradient. We examine the convergence properties of the method and its performance in problems where the simplified level set equation takes the form of a HamiltonJacobi equation with convex/nonconvex Hamiltonian. Importantly, discretizations based on structured and unstructured finiteelement meshes of bilinear quadrilateral and linear triangular elements are shown to perform equally well. Copyright © 2005 John Wiley & Sons, Ltd.
Keywords: finite difference methods;finite element analysis;Galerkin method;gradient methods;nonlinear equations;set theory;


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