arXiv subjects

Carolin Mehlmann

Publications and source records attributed to Carolin Mehlmann.

9 recordsLinked to original sources

arXiv2026

Analysis of a Surface Crouzeix-Raviart Element for the Stokes Problem

Recently, increasing attention has been paid to finite element discretizations of vector-valued flow problems posed on curved surfaces. In this work, we study a surface Stokes system defined on a two-dimensional manifold embedded in three-dimensional space. The surface velocity field is approximated using a nonconforming Crouzeix-Raviart finite element on a polyhedral approximation of the surface, while the pressure is discretized by piecewise constant functions. The governing equations involve the symmetric strain-rate tensor, whose approximation with Crouzeix-Raviart finite elements leads to spurious oscillations in the velocity field because the discrete Korn inequality fails on the nonconforming space. We therefore stabilize the momentum equation by adding an edge-jump penalty term, which restores the coervity of the bilinear term as the discrete Korn inequality holds for this stabilized version. Additionally, we establish that this finite element pair of velocity and pressure spaces satisfies the discrete inf-sup compatibility condition. Furthermore, we derive optimal a-priori error estimates in both the energy norm and the $L^2$-norm. These theoretical results are corroborated by numerical experiments.

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arXiv2026

Analysis and numerical simulations of a landfast ice model

In this manuscript, we consider a common modeling framework for Arctic landfast ice based on the work of Lemieux et al. [27], which is designed for use in large-scale climate models. This approach extends the classical viscous-plastic sea-ice model introduced by Hibler [18], which remains the most used model for simulating large-scale sea-ice dynamics in climate science. In particular, landfast ice refers to sea-ice that is attached to the coastline or grounded and therefore exhibits nearly vanishing motion. We present a rigorous analytical and numerical study of this landfast ice model. The main analytical contributions are the local strong well-posedness, the global strong well-posedness in the absence of external forces and for initial data close to constant equilibrium solutions, and the existence of time-periodic solutions. Complementing the analysis, we perform numerical simulations that illustrate key qualitative differences between landfast ice and classical viscous-plastic sea-ice models. In particular, the simulations reveal the formation of stationary equilibrium states characterized by vanishing ice velocity. These observations are consistent with the global-in-time existence result close to equilibria established in Theorem 4.1 as well as the time-periodic result in Theorem 5.2. The combined analytical and numerical results provide new insight into the structure, stability, and long-term behavior of landfast ice dynamics.

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arXiv2026

A Scalable Monolithic Modified Newton Multigrid Framework for Time-Dependent $p$-Navier-Stokes Flow

Fully implicit tensor-product space-time discretizations of time-dependent $(p,\delta)$-Navier-Stokes models yield, on each time step, large nonlinear monolithic saddle-point systems. In the shear-thinning regime $1 0$ and consistency of the reduced time quadrature. Numerical tests demonstrate robustness with respect to model parameters, nonlinear and linear iteration counts, and scalable parallel performance.

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arXiv2025

A Hybrid Neural Network-Finite Element Method for the Viscous-Plastic Sea-Ice Model

We present an efficient hybrid Neural Network-Finite Element Method (NN-FEM) for solving the viscous-plastic (VP) sea-ice model. The VP model is widely used in climate simulations to represent large-scale sea-ice dynamics. However, the strong nonlinearity introduced by the material law makes VP solvers computationally expensive, with the cost per degree of freedom increasing rapidly under mesh refinement. High spatial resolution is particularly required to capture narrow deformation bands known as linear kinematic features in viscous-plastic models. To improve computational efficiency in simulating such fine-scale deformation features, we propose to enrich coarse-mesh finite element approximations with fine-scale corrections predicted by neural networks trained with high-resolution simulations. The neural network operates locally on small patches of grid elements, which is efficient due to its relatively small size and parallel applicability across grid patches. An advantage of this local approach is that it generalizes well to different right-hand sides and computational domains, since the network operates on small subregions rather than learning details tied to a specific choice of boundary conditions, forcing, or geometry. The numerical examples quantify the runtime and evaluate the error for this hybrid approach with respect to the simulation of sea-ice deformations. Applying the learned network correction enables coarser-grid simulations to achieve qualitatively similar accuracy at approximately 11 times lower computational cost relative to the high-resolution reference simulations. Moreover, the learned correction accelerates the Newton solver by up to 10% compared to runs without the correction at the same mesh resolution.

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arXiv2025

A Hybrid Particle-Continuum Method for Simulating Fast Ice via Subgrid Iceberg Interaction

A significant fraction (4%-13%) of Antarctic sea ice remains stationary as landfast sea-ice ("fast ice"), typically anchored by grounded icebergs. Current global climate models do not represent fast-ice formation due to iceberg grounding, as iceberg-sea-ice interaction mostly occurs at subgrid scales. We propose a novel subgrid-scale coupling mechanism between Lagrangian iceberg particles and an Eulerian sea-ice continuum model. This hybrid particle-continuum approach integrates feedback from icebergs into the sea-ice momentum equation via a Green's function, a Stokeslet, representing the drag exerted by a point force on the viscous-plastic medium. The coupled system, including the Stokeslet induced drag, is discretized using a finite-element method with piecewise linear basis functions. The approach assumes that individual icebergs have diameters smaller than the grid spacing. The presented finite-element discretization is compatible with existing unstructured-mesh ocean model frameworks such as FESOM and ICON, ensuring practical applicability in Earth system modeling. This work provides and analyzes, for the first time, a stable numerical framework to capture the effects of individual subgrid-scale icebergs on sea-ice dynamics. We derive an a-priori stability estimate bounding a functional of the sea-ice system and show that the momentum equation including the subgrid iceberg-sea-ice drag remains stable. Numerical test cases demonstrate the capability of the approach to capture fast-ice formation due to subgrid iceberg grounding on coarse horizontal grids.

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arXiv2023

Analysis of the Crouzeix-Raviart Surface Finite Element Method for vector-valued Laplacians

Recently, a nonconforming surface finite element was developed to discretize 3d vector-valued compressible flow problems arising in climate modeling. In this contribution we derive an error analysis for this approach on a vector-valued Laplace problem, which is an important operator for fluid-equations on the surface. In our setup, the problem is approximated via edge-integration on local flat triangles using the nonconforming linear Crouzeix-Raviart element. The latter is continuous at the midpoints of the edges in each vector component. This setup is numerically efficient and straightforward to implement. For this Crouzeix-Raviart discretization we introduce interpolation estimates, derive optimal error bounds in the H1-norm and L2-norm and present an estimate for the geometric error. Numerical experiments validate the theoretical results.

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arXiv2022

Robust and efficient primal-dual Newton-Krylov solvers for viscous-plastic sea-ice models

We present a Newton-Krylov solver for a viscous-plastic sea-ice model. This constitutive relation is commonly used in climate models to describe the material properties of sea ice. Due to the strong nonlinearity introduced by the material law in the momentum equation, the development of fast, robust and scalable solvers is still a substantial challenge. In this paper, we propose a novel primal-dual Newton linearization for the implicitly-in-time discretized momentum equation. Compared to existing methods, it converges faster and more robustly with respect to mesh refinement, and thus enables numerically converged sea-ice simulations at high resolutions. Combined with an algebraic multigrid-preconditioned Krylov method for the linearized systems, which contain strongly varying coefficients, the resulting solver scales well and can be used in parallel. We present experiments for two challenging test problems and study solver performance for problems with up to 8.4 million spatial unknowns.

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arXiv2021

Sea-ice dynamics on triangular grids

We present a stable discretization of sea-ice dynamics on triangular grids that can straightforwardly be coupled to an ocean model on a triangular grid with Arakawa C-type staggering. The approach is based on a nonconforming finite element framework, namely the Crouzeix-Raviart finite element. As the discretization of the viscous-plastic and elastic-viscous-plastic stress tensor with the Crouzeix-Raviart finite element produces oscillations in the velocity field, we introduce an edge-based stabilization. To show that the stabilized Crouzeix-Raviart approximation is qualitative consistent with the solution of the continuous sea-ice equations, we derive a $H^1$-estimate. In a numerical analysis we show that the stabilization is fundamental to achieve stable approximation of the sea-ice velocity field.

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arXiv2020

A goal oriented error estimator and mesh adaptivity for sea ice simulations

For the first time we introduce an error estimator for the numerical approximation of the equations describing the dynamics of sea ice. The idea of the estimator is to identify different error contributions coming from spatial and temporal discretization as well as from the splitting in time of the ice momentum equations from further parts of the coupled system. The novelty of the error estimator lies in the consideration of the splitting error, which turns out to be dominant with increasing mesh resolution. Errors are measured in user specified functional outputs like the total sea ice extent. The error estimator is based on the dual weighted residual method that asks for the solution of an additional dual problem for obtaining sensitivity information. Estimated errors can be used to validate the accuracy of the solution and, more relevant, to reduce the discretization error by guiding an adaptive algorithm that optimally balances the mesh size and the time step size to increase the efficiency of the simulation.

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