## 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

## 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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### Cite this article

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