Elliptic systems of partial differential equations and the finite element method

An existence and uniqueness theorem is established for finite element solutions of elliptic systems of partial differential equations. To establish this result, an extension of Gårding's inequality is obtained which is valid for functions that do not necessarily vanish on the boundary of the region. To accomplish this extension, a stronger ellipticity condition, called very strong ellipticity, is defined with various necessary and sufficient conditions given.