Traditionally, finite element methods generate progressively higher order accurate solutions by use of higher degree trial space bases for the weak statement construction. This invariably yields matrix equations of greater bandwidth thus increasing implementational and computational costs. A new approach to designing high order, defined here to exceed a third-order accurate method, has been developed and tested. The systematic construction of progressively higher order spatial approximations is achieved via a modified equation analysis, which allows one to clearly identify appended terms appropriate for a desired accuracy order. The resulting "modified" PDE is shown to be consistent with the Taylor Weak Statement (TWS) formulation. It confirms the expected high order of spatial accuracy in TWS constructions and provides a highly efficient dispersion error control mechanism whose application is based on the specifics of the solution domain discretization and physics of the problem. A distinguishing desirable property of the developed method is solution matrix bandwidth containment, i.e., bandwidth always remains equal to that of the linear basis (second-order) discretization. This permits combining the computational efficiency of the lower order methods with superior accuracy inherent in higher order approximations.