Abstract
This study presents a novel goal-oriented error estimate for the nonlinear shallow water equations solved using a mixed discontinuous/continuous Galerkin approach. This error estimator takes account of the discontinuities in the discrete solution and is used to drive two metric-based mesh adaptation algorithms: one which yields isotropic meshes and another which yields anisotropic meshes. An implementation of these goal-oriented mesh adaptation algorithms is described, including a method for approximating the adjoint error term which arises in the error estimate. Results are presented for simulations of two model tidal farm configurations computed using the Thetis coastal ocean model (Kärnä et al. in Geosci Model Dev 11(11):4359–4382, 2018). Convergence analysis indicates that meshes resulting from the goal-oriented adaptation strategies permit accurate QoI estimation using fewer computational resources than uniform refinement.
Highlights
Coastal modelling problems are typically multi-scale, often with a strongly direction-dependent flow
The shallow water equations are solved using the Thetis coastal ocean modelling framework [24], which is based upon the finite element library Firedrake [33]
Where ( h) is the strong PDE residual (3), ( h) concatenates the residuals associated with the boundary conditions and EDK G( h, ∗) contains flux terms arising from the discontinuous Galerkin (DG) discretisation
Summary
Coastal modelling problems are typically multi-scale, often with a strongly direction-dependent flow. The metric based approach to anisotropic mesh adaptation was first introduced in [20] and uses Riemannian metric fields to control the size of mesh elements, and their shape and orientation. This approach was shown to be suited to multi-scale ocean modelling in [31]. This work builds upon the anisotropic goal-oriented mesh adaptation research referenced above, focusing on the shallow water equations. The shallow water equations are solved using the Thetis coastal ocean modelling framework [24], which is based upon the finite element library Firedrake [33]. Application of the error estimation and adaptation techniques to shallow water problems is considered in Sect.
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