An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation
Mathematics of Computation
Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations
Journal of Computational Physics
Numerical schemes for the Hamilton-Jacobi and level set equations on triangulated domains
Journal of Computational Physics
Runge–Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems
Journal of Scientific Computing
hp-Adaptive Discontinuous Galerkin Finite Element Methods for First-Order Hyperbolic Problems
SIAM Journal on Scientific Computing
Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems
SIAM Journal on Numerical Analysis
A hybrid particle level set method for improved interface capturing
Journal of Computational Physics
SIAM Journal on Numerical Analysis
Flux-based level set method: A finite volume method for evolving interfaces
Applied Numerical Mathematics
Implicit tracking for multi-fluid simulations
Journal of Computational Physics
Benchmarking FEniCS for mantle convection simulations
Computers & Geosciences
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In the last two decades, the level set method has been extensively used for the numerical solution of interface problems in different domains. The basic idea is to embed the interface as the level set of a regular function. In this paper we focus on the numerical solution of interface advection equations appearing in free-surface fluid dynamics problems, where naive finite element implementations are unsatisfactory. As a matter of fact, practitioners in fluid dynamics often complain that the mass of each fluid component is not conserved, a phenomenon which is therefore often referred to as mass loss. In this paper we propose and compare two finite element implementations that cure this ill-behaviour without the need to resort to combined strategies (such as e.g. particle level set). The first relies on a discontinuous Galerkin discretization, which is known to give very good performance when facing hyperbolic problems; the second is a stabilized continuous FEM implementation based on the stabilization method presented in [N. Parolini, Computational fluid dynamics for Naval engineering applications, PhD thesis, Ecole Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, June 2004], which is free from many of the problems that classical methods exhibit when applied to unsteady problems.