Preconditioning mixed finite elements for tide models☆
Introduction
Accurate modeling of tides plays an important role in several disciplines. For example, geologists use tide models to help understand sediment transport and coastal flooding, while oceanographers study tides to discern mechanisms sustaining global circulation [1], [2]. Finite element methods making use of unstructured (typically triangular) meshes are attractive to handle irregular coastlines and topography [3]. In many situations, it is sufficient to use a linearized shallow water model with rotation and a parameterized drag term. In particular, the literature contains many papers [4], [5], [6], [7], [8], [9] studying mixed finite element pairs for discretization of each layer for ocean and atmosphere models, and we continue study of this case here.
Much of the literature relates to dispersion relations and enforcement of conservation principles by mixed methods, but our prior work in this area has been to focus on energy estimates. In [10], we gave a careful account of the effect of linear bottom friction in semidiscrete mixed methods, showing that, absent forcing, one obtained exponential damping of a natural energy functional. This allowed estimates of long-time stability and optimal-order a priori error estimates. Then, we handled the (much more delicate) case of a broad family of nonlinear damping terms in [11]. In this case, the energy decay is sub-exponential (typically bounded by a power law) but still strong enough to admit long-time stability and error estimates.
While our work in [10], [11] focused on the semidiscrete mixed finite element case, we now turn to certain issues related to time-stepping. Crank–Nicolson time-stepping is second-order accurate, A-stable (not subject to CFL-like stability condition), and exactly energy conserving in the absence of forcing and damping. However, because it is implicit, it requires the solution of a system of algebraic equations at each time step. For linear damping models, this system is linear, but nonlinear otherwise. The point of this paper is to develop robust preconditioners for the linear system (or Jacobian of the nonlinear one) for use in conjunction with a Krylov method such as GMRES [12].
In addition to the mesh size and time step, our model also depends on a number of physical parameters, described in the following section. Our goal is to design a preconditioner that enables GMRES to converge with an overall iteration counts that depend as little as possible on these parameters. We follow the technique of using weighted-norm preconditioners [13]. Here, one designs an inner product with respect to which the variational problem is bounded with bounded inverse. Such bounds should depend weakly, if at all, on physical and discretization parameters.
The paper is organized as follows. We describe the particular tide model of interest and its discretization in Section 2. This includes Crank–Nicolson time-stepping and a comparison to a symplectic Euler method. Then, we turn to preconditioning the Crank–Nicolson system in Section 3. After analyzing a simple block-diagonal preconditioner with scaled mass matrices, we develop and analyze a parameter-weighted inner product on . Our estimate shows that the preconditioned system has an intrinsic time scale determined by the Rossby number that must be resolved by the time step. This does not seem to be a major practical constraint. After discussion and analysis of these preconditioners, we turn to numerical experiments validating the theory in Section 4 and draw some conclusions in Section 5.
Section snippets
Description of finite element tidal model
The nondimensional linearized rotating shallow water model with linear drag and forcing on a two dimensional surface is given by Here, the unknowns are , which is the nondimensional velocity field tangent to , and is the nondimensional free surface elevation above the height at state of rest. The quantity is just the velocity rotated by . The system is driven by the quantity , which is the (spatially varying) tidal forcing. Several
Preconditioning
Now, we turn to developing a preconditioner for (8). Here, we concretize the abstract approach taken in [21], [22] for our particular tide model. Essentially, a bounded bilinear form on a Hilbert space is equivalent to a linear operator from into its topological dual . Classical Galerkin discretization restricts this bilinear form and operator to some finite-dimensional subspace . Moreover, the discrete operator is encoded by the usual finite element stiffness matrix
Numerical results
We have implemented a mixed finite element discretization of the tide model and developed all of our preconditioners within the Firedrake framework [28]. Firedrake is an automated system for the solution of PDE using the finite element method. It allows users to specify the variational form of their problems using the Unified Form Language (UFL) in Python [29], generates efficient low-level code for the evaluation of operators, and interfaces tightly with PETSc for scalable algebraic solvers.
Conclusions
We have developed effective weighted-norm preconditioners for a mixed finite element/Crank–Nicolson discretization of the linearized rotating shallow water equations with (possibly nonlinear) damping. These preconditioners are based on defining a suitable inner product in which the operators are bounded with bounded inverse in a relatively parameter-independent way. These estimates in turn control the spectrum of the preconditioned operator. Our estimates remain dependent on the ratio ,
CRediT authorship contribution statement
Robert C. Kirby: Conceptualization, Project administration, Funding acquisition, Supervision, Writing. Tate Kernell: Formal analysis, Investigation, Visualization, Writing.
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