# Mixed finite elements for global tide models

## Abstract

We study mixed finite element methods for the linearized rotating shallow water equations with linear drag and forcing terms. By means of a strong energy estimate for an equivalent second-order formulation for the linearized momentum, we prove long-time stability of the system without energy accumulation—the geotryptic state. A priori error estimates for the linearized momentum and free surface elevation are given in \(L^2\) as well as for the time derivative and divergence of the linearized momentum. Numerical results confirm the theoretical results regarding both energy damping and convergence rates.

## Mathematics Subject Classication

65M12 65M60 35Q86## 1 Introduction

Finite element methods are attractive for modelling the world’s oceans since implemention with triangular cells provides a means to accurately represent coastlines and topography [36]. In the last decade or so, there has been much discussion about the best choice of mixed finite element pairs to use as the horizontal discretization for atmosphere and ocean models. In particular, much attention has been paid to the properties of numerical dispersion relations obtained when discretizing the rotating shallow water equations [5, 6, 10, 22, 30, 31, 32, 33]. In this paper we take a different angle, and study the behavior of discretizations of forced-dissipative rotating shallow-water equations, which are used for predicting global barotropic tides. The main point of interest here is whether the discrete solutions approach the correct long-time solution in response to quasi-periodic forcing. In particular, we study the behavior of the linearized energy. Since this energy only controls the divergent part of the solution, as we shall see later, it is important to choose finite element spaces where there is a natural discrete Helmholtz decomposition, and where the Coriolis term projects the divergent and divergence-free components of vector fields correctly onto each other. Hence, we choose to concentrate on the mimetic, or compatible, finite element spaces (i.e. those which arise naturally from the finite element exterior calculus [1]) which were proposed for numerical weather prediction in [7]. In that paper, it was shown that the discrete equations have an exactly steady geostrophic state (a solution in which the Coriolis term balances the pressure gradient) corresponding to each of the divergence-free velocity fields in the finite element space; this approach was extended to develop finite element methods for the nonlinear rotating shallow-water equations on the sphere that can conserve energy, enstrophy and potential vorticity [8, 26, 29]. Here, we shall make use of the discrete Helmholtz decomposition in order to show that mixed finite element discretizations of the forced-dissipative linear rotating shallow-water equations have the correct long-time energy behavior. Since we are studying linear equations, these energy estimates then provide finite time error bounds.

Predicting past and present ocean tides is important because they have a strong impact on sediment transport and coastal flooding, and hence are of interest to geologists. Recently, tides have also received a lot of attention from global oceanographers since breaking internal tides provide a mechanism for vertical mixing of temperature and salinity that might sustain the global ocean circulation [12, 27]. A useful tool for predicting tides are the rotating shallow water equations, which provide a model of the barotropic (i.e., depth-averaged) dynamics of the ocean. When modelling global barotropic tides away from coastlines, the nonlinear advection terms are very weak compared to the drag force, and a standard approach is to solve the linear rotating shallow-water equations with a parameterized drag term to model the effects of bottom friction, as described in [21]. This approach can be used on a global scale to set the boundary conditions for a more complex regional scale model, as was done in [14], for example. Various additional dissipative terms have been proposed to account for other dissipative mechanisms in the barotropic tide, due to baroclinic tides, for example[16].

As mentioned above, finite element methods provide useful discretizations for tidal models since they can be used on unstructured grids which can seamless couple global tide structure with local coastal dynamics. A discontinuous Galerkin approach was developed in [34], whilst continuous finite element approaches have been used in many studies ([11, 18, 23], for example). The lowest order Raviart–Thomas element for velocity combined with \(P_0\) for height was proposed for coastal tidal modeling in [35]; this pair fits into the framework that we discuss in this paper.

- 1.
For the mixed finite element methods that we consider, the spatial semidiscretization also has an attracting solution in the presence of time-varying forcing.

- 2.
This attracting solution converges to time-varying attracting solution of the unapproximated equations.

Although our present work focuses squarely on the shallow water equations, we believe that many of our results will apply to other hyperbolic systems with damping. For example, the model we consider is just the damped acoustic wave equation plus the Coriolis term. Our techniques should extend to other settings where function spaces have discrete Helmholtz decompositions, most notably damped electromagnetics or elastodynamics.

Also, we point out that our main aim here is the theoretical analysis of the damped system. This is the first such analysis of which we are aware demonstrating the strong damping and hence optimal long-time error bounds. We do not assert whether similar results hold for other discretizations, just that they are unkown. While an extended experimental and/or theoretical study of these properties for a wide range of discretizations could be a fruitful project for the tide modelling community, it is beyond the scope of the present work. We do point out that the lowest-order Raviart–Thomas element has been previously proposed for tidal modeling [35], and our framework covers this case as well as the extension to higher-order methods and other mixed spaces.

The rest of this paper is organised as follows. In Sect. 2 we describe the finite element modelling framework which we will analyse. In Sect. 3 we provide some mathematical preliminaries. In Sect. 4 we derive energy stability estimates for the finite element tidal equations. In Sect. 5 we use these energy estimates to obtain error bounds for our numerical solution. Appendix A includes the discussion of embedded manifolds.

## 2 Description of finite element tidal model

*u*is the nondimensional two dimensional velocity field tangent to \({\varOmega }\), \(u^\perp =(-u_2,u_1)\) is the velocity rotated by \(\pi /2\), \(\eta \) is the nondimensional free surface elevation above the height at state of rest, \(\nabla \eta '\) is the (spatially varying) tidal forcing, \(\epsilon \) is the Rossby number (which is small for global tides),

*f*is the spatially-dependent non-dimensional Coriolis parameter which is equal to the sine of the latitude (or which can be approximated by a linear or constant profile for local area models), \(\beta \) is the Burger number (which is also small),

*C*is the (spatially varying) nondimensional drag coefficient and

*H*is the (spatially varying) nondimensional fluid depth at rest, and \(\nabla \) and \(\nabla \cdot \) are the intrinsic gradient and divergence operators on the surface \({\varOmega }\), respectively.

We have already discussed mixed finite elements’ application to tidal models in the geophysical literature, but this work also builds on existing literature for mixed discretization of the acoustic equations. The first such investigation is due to Geveci [13], where exact energy conservation and optimal error estimates are given for the semidiscrete first-order form of the model wave equation. Later analysis [9, 17] considers a second order in time wave equation with an auxillary flux at each time step. In [20], Kirby and Kieu return to the first-order formulation, giving additional estimates beyond [13] and also analyzing the symplectic Euler method for time discretization. From the standpoint of this literature, our model (3) appends additional terms for the Coriolis force and damping to the simple acoustic model. We restrict ourselves to semidiscrete analysis in this work, but pay careful attention the extra terms in our estimates, showing how study of an equivalent second-order equation in \(H(\mathrm {div})\) proves proper long-term behavior of the model.

## 3 Mathematical preliminaries

For the velocity space \(V_h\), we will work with standard \(H(\mathrm {div})\) mixed finite element spaces on triangular elements, such as Raviart–Thomas (RT), Brezzi–Douglas–Marini (BDM), and Brezzi–Douglas–Fortin–Marini (BDFM) [3, 4, 28]. We label the lowest-order Raviart–Thomas space with index \(k=1\), following the ordering used in the finite element exterior calculus [1]. Similarly, the lowest-order Brezzi–Douglas–Fortin–Marini and Brezzi–Douglas–Marini spaces correspond to \(k=1\) as well. We will always take \(W_h\) to consist of piecewise polynomials of degree \(k-1\), not constrained to be continuous between cells. In the case of domains with boundaries, we require the strong boundary condition \(u\cdot n = 0\) on all boundaries.

In the main part of this paper we shall present results assuming that the domain is a subset of \(\mathbb {R}^2\), i.e. flat geometry. In the Appendix, we describe how to extend these results to the case of embedded surfaces in \(\mathbb {R}^3\).

*u*, \(\eta \), and

*h*, although not necessarily of the shapes of the elements in the mesh.

## 4 Energy estimates

**Proposition 1**

**Proposition 2**

In the presence of forcing and dissipation, it is also possible to make estimates showing worst-case linear accumulation of the energy over time.

**Proposition 3**

*F*, we have that for all time

*t*,

*Proof*

*C*is nonzero, we have that

*E*(

*t*) is nonincreasing, although with no particular decay rate.

*v*in (6) be \(v_h = u_{h,t} + \alpha u^D_h\). This gives

*A*(

*t*) and

*B*(

*t*) are comparable to

*E*(

*t*) defined in (28), we can obtain exponential damping of the energy.

**Lemma 1**

*Proof*

Showing that *B*(*t*) is bounded above by a constant times *E*(*t*) is straightforward, but not needed for our damping results.

**Lemma 2**

*Proof*

*B*(

*t*) is exactly \(\alpha E(t)\). However, we are also constrained to pick \(\alpha \le \min \{\alpha _1,\alpha _2\}\) in order to guarantee that the lower bounds for

*A*(

*t*) is positive as well. If we have \(\alpha \le \alpha _2\), then

We combine these two lemmas to give our exponential damping result.

**Theorem 1**

*Proof*

*A*in (37) gives the desired estimate.

*u*to reach its steady state quickly, driving both \(u^D\) and \(u^S\) toward zero at an exponential rate. Finally, since \(\eta _t = -\nabla \cdot u\) almost everywhere, the exponential damping of \(\Vert \nabla \cdot u \Vert \) also forces \(\eta \) toward its zero steady state at the same rate.

**Theorem 2**

*Proof*

*A*(

*t*)

These stability results have important implications for tidal computations. Theorem 2 shows long-time stability of the system. Our stability result also shows that the semidiscrete method captures the three-way geotryptic balance between Coriolis, pressure gradients, and forcing. Moreover, we also can demonstrate that “spin-up”, the process by which in practice tide models are started from an arbitrary initial condition and run until they approach their long-term behavior, is justified for this method. To see this, the difference between any two solutions with equal forcing but differing initial conditions will satisfy the same (6) with nonzero initial conditions and zero forcing. Consequently, the difference must approach zero exponentially fast. This means that we can define a global attracting solution in the standard way [that is, take \(\eta (x,t;t^*)\), \(u(x,t;t^*)\) for \(0 > t^*\) and \(t > t^*\) as the solution starting from zero initial conditions at \(t^*\) and define the global attracting solution as the limit as \(t^* \rightarrow -\infty \)], to which the solution for any condition becomes exponentially close in finite time. The error estimates we demonstrate in the next section then can be used to show that the semidiscrete finite element solution for given initial conditions approximates this global attracting solution arbitrarily well by picking *t* large enough that the difference between the exact solution with those initial conditions and the global attracting solution is small and then letting *h* be small enough that the finite element solution approximates that exact solution well.

## 5 Error estimates

**Proposition 4**

*Proof*

*g*,

**Theorem 3**

Note that our bound on the error equations in Proposition 4 depends only on the approximation properties of the velocity space, while the full error in the finite element solution depends on the approximation properties of both spaces. Consequently, the velocity approximation using BDM elements is suboptimal. Using RT or BDFM elements, both fields are approximated to optimal order.

**Proposition 5**

*Proof*

**Theorem 4**

## 6 Numerical results

In this section we present some numerical experiments that illustrate the estimates derived in the previous sections. In all cases the equations are discretized in time using the implicit midpoint rule. The domain is the unit sphere, centred on the origin, which is approximated using triangular elements arranged in an icosahedral mesh structure (see Appendix A for extensions of the results of this paper to embedded surfaces such as the sphere). All numerical results are obtained using the open source finite element library, Firedrake (http://www.firedrake.org).

*u*are chosen to solve the continuity equation for \(\eta \) exactly, and

*F*is then chosen so that the

*u*equation is satisfied. We used the parameters \(\epsilon =\beta =0.1\), \(f=H=1\), \(C=1000\), \({\varOmega }=2\), and chose \({\varDelta } t=10^{-5}\) in order to isolate the error due to spatial discretization only. We ran the solutions until \(t=0.3\) and computed the time-averaged \(L^2\) error for \(\eta \). Plots are shown in Fig. 2; they confirm the expected first order convergence rate for \(V=\hbox {RT1}\), \(Q=\) DG0, and the expected second order convergence rate for \(V=\hbox {RT2}\), \(Q= \hbox {DG1}\).

*C*and the other parameters (and bounded by \(\alpha \) in Theorem 1). As discussed among our stability results, any two solutions with different initial conditions should converge to the same solution as \(t\rightarrow \infty \). We illustrate this by randomly generating initial conditions for two solutions \((u_1,\eta _1)\) and \((u_2,\eta _2)\) with the same time-periodic forcing,

## 7 Conclusions and future work

We have presented and analyzed mixed finite element methods for the linearized rotating shallow equations with forcing and linear drag terms. Our more delicate energy estimates rely on an equivalence between the first order form and a second order form, and this equivalence itself relies on fundamental properties of classical \(H(\mathrm {div})\) finite elements. In particular, our estimates show that the mixed spatial discretization accurately captures the long-term energy of the system, in which damping balances out forcing to prevent energy accumulation. Because of the linearity of the problem, our energy estimates also give rise to a priori error estimates that are optimal for Raviart–Thomas and Brezzi–Douglas–Fortin–Marini elements. Numerical results confirm both the stability and convergence theory given.

In the future, we hope to extend this work in several directions. First, we hope to study the more realistic quadratic damping model, which will require new techniques to handle the nonlinearity. Second, our estimates have only handled the semidiscrete case, and it is well-known that time-stepping schemes do not always preserve the right energy balances. Without damping or forcing, the implicit midpoint method preserves exact energy balance, and a symplectic Euler method will exactly conserve an approximate functional for linear problems. It remains to be seen how to give a rigorous fully discrete analysis, either including damping by a fractional step or fully implicit method. Finally, even explicit or symplectic time-stepping will require us to consider linear algebraic problems, as it is typically not possible to perform mass lumping for \(H(\mathrm {div})\) spaces on triangular meshes. Implicit methods will require additional care.

## References

- 1.Arnold, D.N., Falk, R.S., Winther, R.: Finite element exterior calculus, homological techniques, and applications. Acta Numer.
**15**(1), 1–155 (2006)MathSciNetCrossRefMATHGoogle Scholar - 2.Arnold, D.N., Falk, R.S., Winther, R.: Finite element exterior calculus: from Hodge theory to numerical stability. Bull. Am. Math. Soc.
**47**(2), 281–354 (2010)MathSciNetCrossRefMATHGoogle Scholar - 3.Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer, New York (1991)CrossRefMATHGoogle Scholar
- 4.Brezzi, F., Douglas Jr, J., Marini, L.D.: Two families of mixed finite elements for second order elliptic problems. Numer. Math.
**47**(2), 217–235 (1985)MathSciNetCrossRefMATHGoogle Scholar - 5.Comblen, R., Lambrechts, J., Remacle, J.F., Legat, V.: Practical evaluation of five partly discontinuous finite element pairs for the non-conservative shallow water equations. Int. J. Numer. Methods Fluids
**63**(6), 701–724 (2010)MathSciNetMATHGoogle Scholar - 6.Cotter, C., Ham, D.: Numerical wave propagation for the triangular P1DG-P2 finite element pair. J. Comput. Phys.
**230**(8), 2806–2820 (2011). doi: 10.1016/j.jcp.2010.12.024 MathSciNetCrossRefMATHGoogle Scholar - 7.Cotter, C.J., Shipton, J.: Mixed finite elements for numerical weather prediction. J. Comput. Phys.
**231**(21), 7076–7091 (2012)MathSciNetCrossRefMATHGoogle Scholar - 8.Cotter, C.J., Thuburn, J.: A finite element exterior calculus framework for the rotating shallow-water equations. J. Comput. Phys.
**257**, 1506–1526 (2014)MathSciNetCrossRefGoogle Scholar - 9.Cowsar, L.C., Dupont, T.F., Wheeler, M.F.: A priori estimates for mixed finite element methods for the wave equation. Comput. Methods Appl. Mech. Eng.
**82**(1–3), 205–222 (1990)MathSciNetCrossRefMATHGoogle Scholar - 10.Danilov, S.: On utility of triangular C-grid type discretization for numerical modeling of large-scale ocean flows. Ocean Dynam.
**60**(6), 1361–1369 (2010)CrossRefGoogle Scholar - 11.Foreman, M., Henry, R., Walters, R., Ballantyne, V.: A finite element model for tides and resonance along the north coast of British Columbia. J. Geophys. Res.: Oceans (1978–2012)
**98**(C2), 2509–2531 (1993)Google Scholar - 12.Garrett, C., Kunze, E.: Internal tide generation in the deep ocean. Annu. Rev. Fluid Mech.
**39**, 57–87 (2007)MathSciNetCrossRefMATHGoogle Scholar - 13.Geveci, T.: On the application of mixed finite element methods to the wave equation. Math. Model. Numer. Anal
**22**, 243–250 (1988)MathSciNetMATHGoogle Scholar - 14.Hill, D., Griffiths, S., Peltier, W., Horton, B., Törnqvist, T.: High-resolution numerical modeling of tides in the western Atlantic, Gulf of Mexico, and Caribbean Sea during the Holocene. J. Geophys. Res.: Oceans (1978–2012)
**116**(C10) (2011)Google Scholar - 15.Holst, M., Stern, A.: Geometric variational crimes: Hilbert complexes, finite element exterior calculus, and problems on hypersurfaces. Found. Comput. Math.
**12**(3), 263–293 (2012)MathSciNetCrossRefMATHGoogle Scholar - 16.Jayne, S.R., Laurent, L.C.S.: Parameterizing tidal dissipation over rough topography. Geophys. Res. Lett.
**28**(5), 811–814 (2001)CrossRefGoogle Scholar - 17.Jenkins, E.W., Rivière, B., Wheeler, M.F.: A priori error estimates for mixed finite element approximations of the acoustic wave equation. SIAM J. Numer. Anal.
**40**(5), 1698–1715 (2002)MathSciNetCrossRefMATHGoogle Scholar - 18.Kawahara, M., Hasegawa, K.: Periodic Galerkin finite element method of tidal flow. Int. J. Numer. Methods Eng.
**12**(1), 115–127 (1978)MathSciNetCrossRefMATHGoogle Scholar - 19.Kinnmark, I.P., Gray, W.G.: Stability and accuracy of spatial approximations for wave equation tidal models. J. Comput. Phys.
**60**(3), 447–466 (1985)CrossRefMATHGoogle Scholar - 20.Kirby, R.C., Kieu, T.T.: Symplectic-mixed finite element approximation of linear acoustic wave equations. Numer. Math.
**130**, 257–291 (2015)MathSciNetCrossRefMATHGoogle Scholar - 21.Lamb, H. (ed.): Hydrodynamics, 6th edn. Dover Publications, New York (1945)MATHGoogle Scholar
- 22.Le Roux, D., Rostand, V., Pouliot, B.: Analysis of numerically induced oscillations in 2D finite-element shallow-water models part I: Inertia-gravity waves. SIAM J. Sci. Comput.
**29**(1), 331–360 (2007)MathSciNetCrossRefMATHGoogle Scholar - 23.Lefevre, F., Lyard, F., Provost, C.L., Schrama, E.J.: FES99: a global tide finite element solution assimilating tide gauge and altimetric information. J. Atmos. Ocean. Technol.
**19**(9), 1345–1356 (2002)CrossRefGoogle Scholar - 24.Leimkuhler, B., Reich, S.: Simulating Hamiltonian Dynamics, vol. 14. Cambridge University Press, Cambridge (2004)MATHGoogle Scholar
- 25.Lynch, D.R., Gray, W.G.: A wave equation model for finite element tidal computations. Comput. Fluids
**7**(3), 207–228 (1979)CrossRefMATHGoogle Scholar - 26.McRae, A.T.T., Cotter, C.J.: Energy-and enstrophy-conserving schemes for the shallow-water equations, based on mimetic finite elements. Q. J. R. Meteorol. Soc.
**40**(168), 2223–2234 (2014)CrossRefGoogle Scholar - 27.Munk, W., Wunsch, C.: Abyssal recipes II: energetics of tidal and wind mixing. Deep-Sea Res. Part I
**45**(12), 1977–2010 (1998)CrossRefGoogle Scholar - 28.Raviart, P.A., Thomas, J.M.: A mixed finite element method for 2nd order elliptic problems. In: Mathematical Aspects of Finite Element Methods (Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975), pp. 292–315. Lecture Notes in Mathematics, vol. 606. Springer, Berlin (1977)Google Scholar
- 29.Rognes, M.E., Ham, D.A., Cotter, C.J., McRae, A.T.T.: Automating the solution of PDEs on the sphere and other manifolds in FEniCS 1.2. Geosci Model Dev Discuss
**6**(3), 3557–3614 (2013)Google Scholar - 30.Rostand, V., Le Roux, D.: Raviart–Thomas and Brezzi–Douglas–Marini finite-element approximations of the shallow-water equations. Int. J. Numer. Methods Fluids
**57**(8), 951–976 (2008)MathSciNetCrossRefMATHGoogle Scholar - 31.Roux, D.Y.L.: Dispersion relation analysis of the \(P^{NC}_1-P_1\) finite-element pair in shallow-water models. SIAM J. Sci. Comput.
**27**(2), 394–414 (2005)MathSciNetCrossRefMATHGoogle Scholar - 32.Roux, D.Y.L.: Spurious inertial oscillations in shallow-water models. J. Comput. Phys.
**231**(24), 7959–7987 (2012)MathSciNetCrossRefMATHGoogle Scholar - 33.Roux, D.Y.L., Pouliot, B.: Analysis of numerically induced oscillations in two-dimensional finite-element shallow-water models part II: Free planetary waves. SIAM J. Sci. Comput.
**30**(4), 1971–1991 (2009)CrossRefGoogle Scholar - 34.Salehipour, H., Stuhne, G., Peltier, W.: A higher order discontinuous Galerkin, global shallow water model: Global ocean tides and aquaplanet benchmarks. Ocean Model.
**69**, 93–107 (2013)CrossRefGoogle Scholar - 35.Walters, R.A.: Coastal ocean models: two useful finite element methods. Cont. Shelf Res.
**25**(7), 775–793 (2005)CrossRefGoogle Scholar - 36.Weller, H., Ringler, T., Piggott, M., Wood, N.: Challenges facing adaptive mesh modeling of the atmosphere and ocean. Bull. Am. Meteorol. Soc.
**91**(1), 105–108 (2010)CrossRefGoogle Scholar

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.