Orthogonal hp-FEM for Elliptic Problems Based on a Non-Affine Concept Pavel ˇ Sol´ ın 1 , Tom´ aˇ s Vejchodsk´ y 2 , and Martin Z´ ıtka 1 1 Department of Mathematical Sciences, University of Texas at El Paso, El Paso, Texas 79968-0514, USA, solin@utep.edu, mzitka@utep.edu 2 Mathematical Institute, Academy of Sciences, ˇ Zitn´a 25, 11567 Praha 1, Czech Republic, vejchod@math.cas.cz Summary. In this paper we propose and test a new non-affine concept of hier- archic higher-order finite elements (hp-FEM) suitable for symmetric linear elliptic problems. The energetic inner product induced by the elliptic operator is used to construct partially orthonormal shape functions which automatically eliminate all internal degrees of freedom from the stiffness matrix. The stiffness matrix becomes smaller and better-conditioned compared to standard types of higher-order shape functions. The orthonormalization algorithm is elementwise local and therefore eas- ily parallelizable. The procedure is extendable to nonsymmetric elliptic problems. Numerical examples including performance comparisons to other popular sets of higher-order shape functions are presented. 1 Introduction and Historical Remarks Hierarchic higher-order finite element methods (hp-FEM) are increasingly popular in computational engineering and science for their excellent approx- imation properties and the potential of reducing the size of finite element models significantly. The resulting discrete problems usually are much smaller compared to standard lowest-order FEM, but they may exhibit rather high condition numbers unless quality higher-order shape functions are used. The concept of the p-FEM as well as the historically first hierarchic higher- order shape functions were introduced by A.G. Peano in the group of B. Szab´o [3] in the mid-1970s. The strong dependence of the condition number of the stiffness and mass matrices on the choice of higher-order shape functions was discovered later, after more advanced p-FEM and hp-FEM computations were performed [2]. In 2001, an affine-equivalent family of well-conditioned hierar- chic finite elements based on integrated Legendre polynomials was proposed by M. Ainsworth and J. Coyle [1]. In this paper we show that the affine concept of finite elements is an ob- stacle on the way to optimal higher-order shape functions. To obtain optimal shape functions, finite elements have to be constructed in the physical mesh.