Mimetic Spectral Element Method for Elliptic Problems

Abstract

Solving Partial Di®erential 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 ¯rst order equations, where one equation is the divergence relation for some conserved quantity. We then rewrite this system of equations in terms of di®erential geometry. The advantages of using di®erential geometry is twofold. The ¯rst 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 de¯ned in these spaces is very well de¯ned. 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. In creating a compatible discretization we have to introduce a type of basis function that does not reconstruct the continuous ¯eld from nodal values at collocation points, but rather reconstruct a continuous ¯eld from integrated values. We call these functions, edge functions, because of their connection with edges rather than nodes. Making use of these edge functions the method shows absolute conservation for arbitrary element order and arbitrarily shaped elements. With a minor adaptation we can extend the method to the more general elliptic type problem, especially anisotropic di®usion problems like Darcy °ow. In both of these cases the use of multiple elements follows quite naturally, and also in the multi-element case we retain theoretical convergence rates and absolute conservation.