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How to stabilize P1NC-P1 element for unsteady waves simulation
Vincent LEGAT
Université catholique de Louvain
Finite Element methods are highly compelling for numerical ocean modeling. On one hand, complex topographic features can faithfully be represented by locally increasing the mesh resolution and because there is no constraint on the mesh topology. On the other hand, the systematic use of local coordinates allows to avoid the classical singularity problem occurring at both poles with structured meshes. Our ocean model uses an efficient mixed finite element pair P1NC-P1 for the primitive shallow-water equations that did not support spurious oscillations (Hanert 2005). This pair is a good compromise between continuous and discontinuous Galerkin methods, and appears to behave rather well for shallow water flows. Moreover, the model consistently conserves mass and tracers (White 2007). In this talk, we firstly address the issue to solve problems on the sphere (and even on any curved geometries, in a more general sense). Any global coordinates system cannot be used, since it introduces poles and will generate singularity for the representation of all fields at poles. In a second part, we also adress the issue to efficiently and accurately the element pair P1NC-P1 where the compromise between continous and discontinous Galerkin methods does not allow to introduce a straighforward application of the usual approach used in DG methods. However, it is possible to use the same ideas to build an efficient method. We present some validation results with the benchmark test cases described by Williamson et al. (1992). It consists on idealized, but quite realistic non-viscous flows on the sphere. We are able to circumvent the singularity problems inherent to global coordinate systems typically encountered. We also demonstrate that accurate and stable results can be obtained with the stabilized version of the method. This new formulation appears to be quite more robust, stable and accurate than the previous implementations of the mixed finite element pair P1NC-P1.
Date: 2007-12-14 à 05:30
Endroit: Pavillon Adrien-Pouliot, local 2548