INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 2008; 73:1374–1394 Published online 9 July 2007 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nme.2129 Higher-order finite elements based on generalized eigenfunctions of the Laplacian Pavel ˇ Sol´ ın 1, 2, ∗, † and Tom´ aˇ s Vejchodsk´ y 3 1 Institute of Thermomechanics, Academy of Sciences of the Czech Republic, Dolejˇ skova 5, CZ-18200 Prague, Czech Republic 2 Department of Mathematical Sciences, University of Texas at El Paso, 500 West University, El Paso, TX 79968, U.S.A. 3 Mathematical Institute, Academy of Sciences of the Czech Republic, Prague, Czech Republic SUMMARY We present a new class of higher-order finite elements based on generalized eigenfunctions of the Laplace operator, which are suitable for both product and simplicial geometries in R d . Due to simultaneous orthogonality of the generalized eigenfunctions under both the H 1 0 and L 2 products and their almost negligible dependence on reference maps, such finite elements are an excellent choice for the discretiza- tion of second-order elliptic problems by the hp-FEM. Analysis is illustrated by numerical results and comparisons with other popular higher-order finite elements are presented. The new elements are used to compute efficiently the model of an electrostatic micromotor. Copyright 2007 John Wiley & Sons, Ltd. Received 6 September 2006; Revised 10 April 2007; Accepted 23 May 2007 KEY WORDS: hp-FEM; optimal shape functions; generalized eigenfunctions; electrostatic micromotor ∗ Correspondence to: Pavel ˇ Sol´ ın, Department of Mathematical Sciences, University of Texas at El Paso, 500 West University, El Paso, TX 79968, U.S.A. † E-mail: solin@utep.edu Contract/grant sponsor: Grant Agency of the Czech Republic; contract/grant numbers: 102/05/0629, 102/07/0496, 201/04/P021 Contract/grant sponsor: Grant Agency of the Academy of Sciences of the Czech Republic; contract/grant number: IAA100760702 Contract/grant sponsor: U.S. Department of Defense; contract/grant number: 05PR07548-00 Contract/grant sponsor: Academy of Sciences of the Czech Republic; contract/grant number: AV0Z10190503 Copyright 2007 John Wiley & Sons, Ltd.