The penalty method for grid matching in mixed finite element methods

Possibility of use of the penalty method of grid matching for mixed finite element methods is proved in the article. Hermann–Johnson scheme for biharmonic equation is considered in the article. The main idea is to build perturbed problem with two parameters, which play roles of penalties. Perturbed problem is built by substituting essential conditions in mixed variational formulation by natural conditions, which contain parameters, on interface. Discretization of perturbed problem by finite element method is made. Estimations of norm of difference between solution of discrete perturbed problem and solution of initial problem, which depend on step and penalty, are derived. Recommendations to choose penalties in dependence on step are given.