Abstract

The main goal of this lecture is to discuss the numerical solution of the Dirichlet problem for the Eikonal equation

(EKN)                               |nabla u| = 1 in Omega;       u=g on partial Omega

where Omega is a bounded domain of \bf{R}^2. Traditionally, the equation is considered as a nonlinear hyperbolic one leading to numerical methods taking advantage of this hyperbolic feature. Motivated by the numerical solution of systems of the form

nabla \bf{u}(x) in O(d),

completed by boundary conditions (with O(d) the group of the d x d orthogonal matrices), a problem considered by B. Dacorogna, we have investigated the solution of (EKN) through the use of finite elements and elliptic solvers. The main idea is to approximate (EKN) by a problem of Calculus of Variations involving a Ginzburg-Landau nonlinearity and a bi-harmonic regularization (linear or nonlinear). Next, a well-chosen initial value problem associated with the Euler-Lagrange equation of the above variational problem is solved by an operator-splitting method decoupling differential operators and nonlinearity. Assuming that (EKN) has multiple solutions, the above approach allows to compute at will the maximal or the minimal solution of (EKN). This approach is easy to implement via piecewise linear approximations, ideally suited to capture the solutions of this problem, and as such is easy to implement for domains Omega of arbitrary shape, including therefore curved boundaries. The results of numerical experiments will be presented.