Energy estimate for inhomogeneous boundary data

Establish an energy estimate for the discontinuous Galerkin discretization of the Kelvin–Voigt damped intrinsic beam equations with inhomogeneous velocity and sectional-resultant boundary data, despite the sign-indefinite boundary-work terms in the discrete energy balance.

Background

The semi-discrete energy identity proved in the paper assumes homogeneous boundary data. For general inhomogeneous boundary data, the characteristic outer-state construction generates terms that are linear in the prescribed data and coupled to boundary traces of the numerical solution. The authors state that these terms are sign-indefinite and cannot be absorbed into the available dissipative terms by completing the square. Consequently, the paper does not provide an energy estimate for this inhomogeneous closure.