A Penalty Method for Some Nonlinear Variational Obstacle Problems∗

We formulate a penalty method for the obstacle problem associated with a nonlinear variational principle. It is proven that the solution to the relaxed variational problem (in both the continuous and discrete settings) is exact for finite parameter values above some calculable quantity. To solve the relaxed variational problem, an accelerated forward-backward method is used, which ensures convergence of the iterates, even when the Euler-Lagrange equation is degenerate and nondifferentiable. Several nonlinear examples are presented, including quasi-linear equations, degenerate and singular elliptic operators, discontinuous obstacles, and a nonlinear two-phase membrane problem.