An interior penalty method for a large-scale finite-dimensional nonlinear double obstacle problem

Abstract

• Novel interior penalty formulation for large-scale bounded NCPs in engineering. • Proof of unique solvability of the penalty equation. • Establishment of convergence theory for the penalty method. • Jacobian matrix in Newton method shown to be positive-definite and an M-matrix. • Linearized system in Newton method decomposable into two decoupled subsystems. We propose and analyze an interior penalty method for a finite-dimensional large-scale bounded Nonlinear Complementarity Problem (NCP) arising from the discretization of a differential double obstacle problem in engineering. Our approach is to approximate the bounded NCP by a nonlinear algebraic equation containing a penalty function with a penalty parameter μ > 0. The penalty equation is shown to be uniquely solvable. We also prove that the solution to the penalty equation converges to the exact one at the rate O ( μ 1 / 2 ) as μ → 0. A smooth Newton method is proposed for solving the penalty equation and it is shown that the linearized system is reducible to two decoupled subsystems. Numerical experiments, performed on some non-trivial test examples, demonstrate the computed rate of convergence matches the theoretical one.