COMPARISON OF H AND P FINITE-ELEMENT APPROXIMATIONS OF THE SHALLOW-WATER EQUATIONS

A p-type finite element scheme is introduced for the three-dimensional shallow water equations with a harmonic expansion in time. The wave c ontinuity equation formulation is used which decouples the problem int o a Helmholtz equation for surface elevation and a momentum equation f or horizontal velocity. An exploration of the applicability of p metho ds to this form of the shallow water problem is presented, with a cons ideration of the problem of continuity errors. The convergence rates a nd relative computational efficiency between h- and p-type methods are compared with the use of three test cases representing various degree s of difficulty. A channel test case establishes convergence rates, a continental shelf test case examines a problem with accuracy difficult ies at the shelf break, and a field-scale test case examines problems with highly irregular grids. For the irregular grids, adaptive h combi ned with uniform p refinement was necessary to retain high convergence rates.