Note: The solution is expressed in terms of sum of products of Legendre polynomials inside each triangle, independently from neighbouring triangles. A triangular truncation is applied to the base of basis functions.
A triangular truncation reduces the number of unknowns in the problem and does not affect the solution. The link between elements is done through a boundary integral for which the value is taken to be the mean of the two values inside each neighbouring elements. A obvious limitation of the method is therefore that the integration in time has to be explicit. We favored a Runge-Kutta 4th order. In elements sharing one face with a wall, velocities are rotated and the normal component is projected, and hence is zero at the wall. This ensures the no normal flow condition. This can also be done to the element sharing only one vertex with the wall, but does not seem necessary in our case. An important remark concerns the stability of the code. The method is free of numerical noise although nothing special was done to lower the number of degrees of freedom (cf: Iskandarani et al. lowered the number of freedom for the pressure).
Note: This slide shows the accuracy of one finite element model compared to the spectral element model (SPOC) with respect to the resolution (numbers of nodes in the grid (top) and numbers of degrees of freedom (bottom). The number of degrees of freedom for a conventional FEM is the number of nodes. The number of degrees of freedom for SPOC is the number of elements multiplied by the number of degrees of freedom inside each triangle (nc=1=>3, nc=2=>6, nc=3=>10, nc=4=>15, nc=5=>21, nc=6=>28, nc=7=>36, ...). The test is very simple. A wave inside a sqare propagates only in one direction. There should be no velocity component in the cross-direction. But there is in the actual results because of the irregularity of the grids. The slope is directly related to the precision of the models. Low, almost zero for a conventional FEM, interesting for SPOC when nc>5).
The ability of SPOC to represent advection was tested using the advection of a gaussian hill by a simple body rotation. For nc=5, the ability is lower than expected, but nc=7 is ok. Nethertheless, nc=5 is the value used in the following applications.
the test of the geostrophic eddy with no dissipation and no forcing was used to show the ability of SPOC to conserve properties such as energy. In this case, SPOC does as well as the FDM model (the two flat lines). For comparison, the other FEM does a poor job.
Two comparisons between SPOC, the FDM and one stable but dissipative FEM. The energy curves (left panels) during spin-up for SPOC and the FDM are on top of each other. Except for a 2% error in the last case at the end of the simulation. 2% is inside the error bars due to different resolutions.
The conclusion is therefore that SPOC has been tested and validated against a reference FDM (C-grid plus Runge Kutta 4th in time). Further applications will follow.