A penalty/Newton/conjugate gradient method for the solution of obstacle problems

Motivated by the search for non-negative solutions of a system of Eikonal equations with Dirichlet boundary conditions, we discuss in this Note a method for the numerical solution of parabolic variational inequality problems for convex sets such as $\text{K={v∣v∈H}{}_0{}^1\text{(Ω)}$, v?ψ a.e. on $\text{Ω}.}$ The numerical methodology combines penalty and Newton's method, the linearized problems being solved by a conjugate gradient algorithm requiring at each iteration the solution of a linear problem for a discrete analogue of the elliptic operator I?μΔ. Numerical experiments show that the resulting method has good convergence properties, even for small values of the penalty parameter. To cite this article: R. Glowinski et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).