Solution Adaptive Triangular Meshes with Application to the Simulation of Plasma Equilibrium.

Triangular meshes offer a great deal of flexibility in the presence of complicated geometrical configurations and changing topologies. Their general connectivity properties allows the topology of the mesh to change while the vertex coordinates remain invariant in physical space. When coupled to an adaptive strategy, irregular triangular meshes become a powerful tool. We construct a new discrete laplacian operator on a local mesh molecule, second order accurate on symmetric cell regions, based on local Taylor series expansions. This discrete laplacian is then compared to the one commonly used in the literature. A truncation error analysis of gradient and laplacian operators calculated at triangle centroids reveals that the maximum bounds of their truncation errors are minimized on equilateral triangles, for a fixed triangle perimeter. A new adaptive strategy on arbitrary triangular grids is developed in which a uniform grid is defined with respect to the solution surface, as opposed to the x,y plane. Departures from mesh uniformity arises from a spacially dependent mean-curvature of the solution surface. By pulling the nodes on an abstract solution surface that encompasses all the salient features of interest, we are able to eliminate the free parameter usually associated with gradient pull. The power of this new adaptive technique is applied to the problem of finding free-boundary plasma equilibria within the context of MHD. The geometry is toroidal, and axisymmetry in the toroidal direction is assumed. We are led to conclude that the grid should move, not towards regions of high curvature of magnetic flux, but rather towards regions of greater toroidal current density. This has a direct bearing on the accuracy with which the Grad -Shafranov equation is being approximated.