We present a construction of high order finite elements for H1, H(curl), H(div) (and L2) on a pyramid, which are compatible with existing tetrahedral and hexahedral high order finite elements and satisfy the commuting diagram property.

Two related approaches for solving linear systems that arise from a higher-order finite element discretization of elliptic partial differential equations are described. The first approach explores direct application of an algebraic-based multigrid method (AMG) to iteratively solve the linear systems that result from higher-order discretizations. While the choice of basis used on the discretizat...

Low order edge elements are widely used for electromagnetic field problems. Higher order edge approximations are receiving increasing interest but their definition become rather complex. In this paper we propose a simple definition for Whitney edge elements of polynomial degree higher than one. We give a geometrical localization of all degrees of freedom over particular edges and provide a basi...

The paper discusses algebraic multigrid (AMG) methods for the solution of large sparse linear systems arising from the discretization of scalar elliptic partial differential equations with Lagrangian finite elements of order at most 4. The resulting system matrices do not have the M-matrix property that is used by standard analyzes of classical AMG and aggregation-based AMG methods. A unified a...

Every finite field has many multiplicative generators. However, finding one in polynomial time is an important open problem. In fact, even finding elements of high order has not been solved satisfactorily. In this paper, we present an algorithm that for any positive integer c and prime power q, finding an element of order exp(Ω( √ qc)) in the finite field Fq(qc−1)/(q−1) in deterministic time (q...