Carolin Mehlmann: A Surface Crouzeix-Raviart Element for Geophysical Flow Problems

Abstract:

This talk addresses the mathematical challenges and numerical treatment of large-scale sea-ice dynamics in climate models. The underlying model adopts a viscous-plastic framework to describe sea-ice as a two-dimensional thin layer on the ocean surface. In the first part of the talk, we explore the formulation of the model, focusing on developing a presentation that is well-suited for modern numerical approximation techniques. In the second part we will focus on numerical tools for an approximation of the model with finite elements. Specifically, we present a new approach which approximates the flow via edge integration on local flat triangles using the nonconforming linear Crouzeix-Raviart element. This discretization is implemented in the ICON weather and climate model [3]. To address oscillations in the velocity field caused by the discretization of the viscous-plastic stress tensor with the Crouzeix-Raviart element, we propose an edge-based stabilization technique [1]. On the meshes used in climate models the Coruzeix-Raviart discretization is a good compromise between accuracy and efficiency of the numerical setup [2]. In the final part of the talk, we derive optimal error bounds for the Crouzeix-Raviart element in both the H1-norm and L2-norm. These theoretical findings are validated by numerical experiments [4].

[1] C. Mehlmann, P. Korn: Sea-ice dynamics on triangular grids, J. of Computational Physics, 428, eid 110086, 2021 https://doi.org/10.1016/j.jcp.2020.110086

[2] C. Mehlmann, S. Danilov, L. Losch, J.F. Lemieux, P. Blain, C. Hunke, P. Korn: Simulating linear kinematic features in viscous-plastic sea ice models on quadrilateral and triangular grids, Journal of Advanced Earth System Modeling, 13, e2021MS002523, 2021 https://doi.org/10.1029/2021MS002523

[3] C. Mehlmann, O. Gutjahr: Modeling sea-ice dynamics in the tangential plane on the sphere, Journal of Advances in Modeling Earth Systems, 14, e2022MS003010,2022 https://doi.org/10.1029/2022MS003010

[4] C. Mehlmann: Analysis of the Crouzeix-Raviart Surface Finite Element Method for vector-valued Laplacians, e2312.16541, arXiv, 2024, https://arxiv.org/abs/2312.16541