A power penalty method for solving a nonlinear parabolic complementarity problem

Abstract In this paper we present a penalty method for solving a complementarity problem involving a second-order nonlinear parabolic differential operator. 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 e . 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 + e ( 1 + λ e 1 / 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.