The penalty method of grids matching in the mixed Herrmann–Miyoshi scheme

The penalty method for mixed finite element methods is formulated and studied. The Herrmann–Miyoshi scheme for the biharmonic equation is considered. The main idea is to construct a perturbation problem with two parameters that play the role of penalties. The perturbation problem is constructed by substitution of the main conditions in the mixed variational formulation on the interface by natural conditions that contain parameters. The discretization of the perturbation problem by the finite element method is done. Estimates of the norm of a difference between the solution of a discrete perturbation problem and that of a given problem are obtained. Recommendations for choosing penalties depending on a mesh size and penalties are given.