Communications in Mathematical Sciences
Volume 16 (2018)
A penalty method for some nonlinear variational obstacle problems
Pages: 1757 – 1777
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.
nonlinear obstacle problem, sparsity, penalty method, variational methods
2010 Mathematics Subject Classification
Primary 35A15. Secondary 35J60, 35R35.
H.S. acknowledges the support of AFOSR, FA9550-17-1-0125.
Received 9 January 2017
Accepted 20 March 2018