Mimetic Spectral Element Method for Elliptic Problems

Solving Partial Differential Equations (PDE's) numerically requires that the PDE or system of PDE's be replaced with a system of algebraic equations. The replacing system of algebraic equation should be mimetic in the sense that discrete operators that make up the PDE mimic the vector identities that connect the continuous operators. The equation that we focus on is the Poisson equation. The Poisson equation can be split up into two first order equations, where one equation is the divergence relation for some conserved quantity. We then rewrite this system of equations in terms of differential geometry. The advantages of using differential geometry is twofold. The first is that there is a very obvious link to its discrete counterpart which is algebraic topology. Second is that the mapping of spaces and the functions defined in these spaces is very well defined. With these properties we are able to derive a compatible (mimetic) discretization for the Poisson equation for arbitrarily shaped curved spectral elements. On these curved elements we are able to retain exponential convergence.Aerospace Engineerin