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A penalty-regularization-operator splitting method for the numerical solution of a scalar Eikonal equation

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Abstract

In this article, we discuss a numerical method for the computation of the minimal and maximal solutions of a steady scalar Eikonal equation. This method relies on a penalty treatment of the nonlinearity, a biharmonic regularization of the resulting variational problem, and the time discretization by operator-splitting of an initial value problem associated with the Euler-Lagrange equations of the regularized variational problem. A low-order finite element discretization is advocated since it is well-suited to the low regularity of the solutions. Numerical experiments show that the method sketched above can capture efficiently the extremal solutions of various two-dimensional test problems and that it has also the ability of handling easily domains with curved boundaries.

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Author information

Correspondence to Alexandre Caboussat.

Additional information

In Honor of the Scientific Contributions of Professor Luc Tartar

This work was supported by the National Science Foundation (No.DMS-0913982).

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Caboussat, A., Glowinski, R. A penalty-regularization-operator splitting method for the numerical solution of a scalar Eikonal equation. Chin. Ann. Math. Ser. B 36, 659–688 (2015). https://doi.org/10.1007/s11401-015-0930-8

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Keywords

  • Eikonal equation
  • Minimal and maximal solutions
  • Regularization methods
  • Penalization of equality constraints
  • Dynamical flow
  • Operator splitting
  • Finite element methods

2000 MR Subject Classification

  • 65N30
  • 65K10
  • 65M60
  • 49M20
  • 35F30