A power penalty method for a nonlinear parabolic complementarity problem

In this paper we present a penalty method for solving a complementarity problem involving 2nd-order nonlinear parabolic differential operators. In this work we first rewrite the complementarity problem as a nonlinear variational inequality. Then, we define a nonlinear parabolic partial differential equation (PDE) approximating the variational inequality using a power penalty term with a penalty constant λ > 1, a power parameter k > 0 and a smoothing parameter ε. We prove that the solution to the penalized PDE converges to that of the variational inequality in an appropriate norm at an arbitrary exponential rate of the form O([λ−k + ε(1 + λε1/k)]1/2). Numerical experiments, performed to verify the theoretical results, show that the computed rates of convergence in both λ and k are close to the theoretical ones.