Discontinuous Galerkin Finite Volume Element Methods for Second-Order Linear Elliptic Problems Sarvesh Kumar, Neela Nataraj, Amiya K. Pani Industrial Mathematics Group, Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India Received 3 October 2007; accepted 20 August 2008 Published online 30 April 2009 in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/num.20405 In this article, a one parameter family of discontinuous Galerkin finite volume element methods for approx- imating the solution of a class of second-order linear elliptic problems is discussed. Optimal error estimates in L 2 and broken H 1 - norms are derived. Numerical results confirm the theoretical order of convergences. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 25: 1402–1424, 2009 Keywords: discontinuous Galerkin methods; finite volume element methods; optimal error estimates I. INTRODUCTION In this article, we formulate and analyze a one parameter family of discontinuous Galerkin finite volume element methods (DGFVEM) for approximating the solution of the following elliptic problem: Given f , find u such that −∇ · (K ∇u) = f in , (1.1) u = 0 on ∂, where  is a bounded, convex polygonal domain in R 2 with boundary ∂ and K = (k ij (x)) 2×2 is a real valued, symmetric and uniformly positive definite matrix, i.e., there exists a positive constant α 0 such that ξ T Kξ ≥ α 0 ξ T ξ ∀ξ ∈ R 2 . (1.2) In recent years, there has been a renewed interest in Discontinuous Galerkin (DG) methods for the numerical approximation of partial differential equations. This is due to their flexibility in local mesh adaptivity and handling nonuniform degrees of approximation for solutions whose smoothness exhibit variation over the computational domain. DG methods have an advantage Correspondence to: Neela Nataraj, Industrial Mathematics Group, Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India (e-mail: nataraj.neela@gmail.com) © 2009 Wiley Periodicals, Inc.