Spectral Element Approximation of the Hodge-⋆ Operator in Curved Elements

Abstract

Mimetic approaches to the solution of partial differential equations (PDE’s) produce numerical schemes which are compatible with the structural properties – conservation of certain quantities and symmetries, for example – of the systems being modelled. Least Squares (LS) schemes offer many desirable properties, most notably the fact that they lead to symmetric positive definite algebraic systems, which represent an advantage in terms of computational efficiency of the scheme. Nevertheless, LS methods are known to lack proper conservation properties which means that a mimetic formulation of LS, which guarantees the conservation properties, is of great importance. In the present work, the LS approach appears in order to minimize the error between the dual variables, implementing weakly the material laws, obtaining an optimal approximation for both variables. The application to a 2D Poisson problem and a comparison will be made with a standard LS finite element scheme, see, for example, Cai et al. (J. Numer. Anal. 34:425–454, 1997).