A local space–time discontinuous finite element method Jack Chessa a , Ted Belytschko b, * a Department of Mechanical and Industrial Engineering, The University of Texas, El Paso, TX 79968-0521, USA b Department of Mechanical Engineering, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208-3111, USA Received 24 October 2004; received in revised form 21 March 2005; accepted 9 May 2005 Abstract We present a locally enriched space–time finite element method for solving hyperbolic problems with discontinuities. The space–time formulation is coupled to a standard semidiscrete finite element method. The discontinuities are cap- tured by a space–time version of the extended finite element method, which treats arbitrary moving discontinuities. Since the discontinuities are local, the enriched space–time method is only needed around the discontinuities, which pro- vides significant computational savings. The coupling is implemented through a weak enforcement of the continuity of the flux between the space–time and semidiscrete domains in a manner similar to discontinuous Galerkin methods. The method is illustrated through one-dimensional problems. It displays remarkable accuracy in capturing moving discon- tinuities, both in their amplitude and speed. Ó 2005 Elsevier B.V. All rights reserved. Keywords: Enriched; Finite element; Space–time; X-FEM 1. Introduction Enriched finite element methods are capable of reproducing discontinuous features and regions of high gradients without significant mesh refinement. This is accomplished by extending the finite element approx- imation space to be able to reproduce specific enrichment functions that are representative of these features [1]. The enrichment is typically facilitated by the partition-of-unity property of standard Lagrange interpo- lants [2]. These methods have proven quite successful in modeling static and quasi-static phenomena such 0045-7825/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.cma.2005.05.022 * Corresponding author. Tel.: +1 847 491 4029; fax: +1 847 491 4011. E-mail addresses: jfchessa@utep.edu (J. Chessa), tedbelytschko@northwestern.edu, t-belytschko@northwestern.edu (T. Belytschko). Comput. Methods Appl. Mech. Engrg. 195 (2006) 1325–1343 www.elsevier.com/locate/cma