Finite element methodologies have long been used in modeling
circulation in coastal and oceanic waters.  Many codes in wide use
today, e.g., the QUODDY and ADCIRC simulators, are based on continuous
Galerkin (CG) finite elements for both elevation and momentum.
Because continuous elements applied to the primitive continuity
equation are subject to spurious oscillations and/or phase errors,
these codes utilize a reformulation of the continuity equation into a
so-called generalized wave continuity equation (GWCE).  Moreover,
these finite element codes are not ``locally conservative,'' that is,
they do not produce mass conservative fluxes.  In recent years, finite
element methods based on discontinuous basis functions have been
developed (the discontinuous Galerkin or DG method).  These methods
are locally conservative and generally provide sharper resolution of
fronts with minimal oscillation and phase error.  However, these
methods can involve more degrees of freedom than their continuous
counterparts.  In this talk, we will present a multi-algorithmic
finite element strategy based on utilizing both continuous and
discontinuous Galerkin methods. For example, given a system of coupled
equations, we may choose to solve some of them using DG and some using
CG methods, or we may couple CG/DG within the same equation through
boundary conditions.  We will focus on recent developments in the
ADCIRC simulator, where we have replaced the GWCE CG formulation with
a DG formulation based on the primitive continuity equation.
Numerical results will be presented for this new approach.  We will
also discuss the recent development and analysis of a coupled CG/DG
formulation for continuity based on coupling the GWCE with the
primitive continuity equation.