The penalty method for grid matching in mixed finite element methods

In this paper we prove the possibility of the use of the penalty method for grid matching in mixed finite element methods. We consider the Hermann-Johnson scheme for biharmonic equation. The main idea is to construct a perturbed problem with two parameters which play roles of penalties. The perturbed problem is built by the replacement of essential conditions on the interface in the mixed variational statement with natural conditions that contain parameters. The perturbed problem is discretized by the finite element method. We estimate the norm of the difference between a solution of the discrete perturbed problem and a solution of the initial problem; the obtained estimates depend on the step and the penalties. We give recommendations for the choice of penalties depending on the step.