Basis functions for concave polygons

AbstractPolynomials suffice as finite element basis functions for triangles, parallelograms, and some other elements of little practical importance. Rational basis functions extend the range of allowed elements to the much wider class of well-set algebraic elements, where well-set is a convexity type constraint. The extension field from R(x,y) to R(x,y,x2+y2) removes this quadrilateral constraint as described in Chapter 8 of [E.L. Wachspress, A Rational Finite Element Basis, Academic Press, 1975]. The basis function construction described there is clarified here, first for concave quadrilaterals and then for concave polygons. Its application is enhanced by the GADJ algorithm [G. Dasgupta, E.L. Wachspress, The adjoint for an algebraic finite element, Computers and Mathematics with Applications, doi:10.1016/j.camwa.2004.03.021] for finding the denominator polynomial common to all the basis functions

Minuscule representations, invariant polynomials, and spectral covers

Given a minuscule representation of a simple Lie algebra, we find an algebraic model for the action of a regular element and show that these models can be glued together over the adjoint quotient, viewed as the set of all regular conjugacy classes of the Lie algebra. There are partial results in the case of a quasiminuscule representation, and a conjecture in the case of a general irreducible finite-dimensional representation. The method of proof is to relate the question to a problem concerning holomorphic principal bundles over cuspidal cubic curves.Comment: LaTeX, 42 pages, final version, to appear in the proceedings of the University of Missouri conference on Hilbert schemes, vector bundles and representation theory, new material on extensions and the adjoint representation of a simply laced Lie algebra adde

Adjoint-Based Error Estimation and Mesh Adaptation for Hybridized Discontinuous Galerkin Methods

We present a robust and efficient target-based mesh adaptation methodology, building on hybridized discontinuous Galerkin schemes for (nonlinear) convection-diffusion problems, including the compressible Euler and Navier-Stokes equations. Hybridization of finite element discretizations has the main advantage, that the resulting set of algebraic equations has globally coupled degrees of freedom only on the skeleton of the computational mesh. Consequently, solving for these degrees of freedom involves the solution of a potentially much smaller system. This not only reduces storage requirements, but also allows for a faster solution with iterative solvers. The mesh adaptation is driven by an error estimate obtained via a discrete adjoint approach. Furthermore, the computed target functional can be corrected with this error estimate to obtain an even more accurate value. The aim of this paper is twofold: Firstly, to show the superiority of adjoint-based mesh adaptation over uniform and residual-based mesh refinement, and secondly to investigate the efficiency of the global error estimate

Algebraic Anosov actions of Nilpotent Lie groups

In this paper we classify algebraic Anosov actions of nilpotent Lie groups on closed manifolds, extending the previous results by P. Tomter. We show that they are all nil-suspensions over either suspensions of Anosov actions of Z^k on nilmanifolds, or (modified) Weyl chamber actions. We check the validity of the generalized Verjovsky conjecture in this algebraic context. We also point out an intimate relation between algebraic Anosov actions and Cartan subalgebras in general real Lie groups.Comment: 40 page