L2 error estimation for DGFEM for elliptic problems with low regularity

This paper presents an error estimate in the L2-norm for the discontinuous Galerkin finite element methods (DGFEM) for elliptic problems with low regularity solutions. The Raviart–Thomas interpolation operator is employed to derive the new result, which complements the mesh-dependent energy norm error estimates in Gudi (2010) [2]. Numerical results corroborate the theoretical analysis. © 2012 Elsevier Ltd. All rights reserved. 1. DGFEM for elliptic problems The discontinuous Galerkin (DG) finite element methods are widely used in scientific computing and engineering applications. However, the standard a priori error analysis of DGFEM requires additional regularity on solutions. In particular, for second order elliptic problems, it is usually assumed [1] that the solutions are inH1+s, s > 1/2. Recently, there have been efforts on analyzing DGFEM for the problems with low regularity solutions. A priori error estimates in the mesh-dependent energy norms are derived in [2] by applying new techniques that incorporate ideas usually seen in a posteriori analysis. Theoretical estimates and numerical results on DGFEM for elliptic problems with solutions in W 2,p, p < 2 can be found in [3]. The purpose of this paper is to provide an error estimate in the L2-norm for DGFEM for elliptic problems with low regularity solutions. Our approach is simpler than the one used in [4]. Numerical results are presented to illustrate the theoretical analysis. We consider the following model elliptic boundary value problem ∇ · (−K∇u) = f in Ω, u = 0 on ∂Ω, (1) where Ω ⊂ R2 is a bounded polygon and K is a symmetric positive-definite permeability tensor. We adopt the standard definitions for the Sobolev spaces Hs(D) and their associated inner products (·, ·)s,D, norms ∥ · ∥s,D, and seminorms | · |s,D for s ≥ 0. The space H0(D) coincides with L2(D), for which the norm and inner product are denoted as ∥ · ∥D and (·, ·)D, respectively. If D = Ω , we drop D. Let Th be a regular triangular mesh on Ω , Eh the set of all edges in Th, E i h the set of all interior edges. We adopt the definition in [5] for the broken Sobolev space H1(Ω, Th) and define a DG finite element space Vh = {v ∈ L2(Ω): v|T ∈ Pk(T ), ∀T ∈ Th}, (2) ∗ Corresponding author. Tel.: +1 970 491 3067. E-mail addresses: [email protected] (J. Liu), [email protected] (L. Mu), [email protected] (X. Ye). 0893-9659/$ – see front matter© 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.aml.2012.01.022 J. Liu et al. / Applied Mathematics Letters 25 (2012) 1614–1618 1615 where k ≥ 1 is the degree of polynomial shape functions. We adopt also the standard definitions for averages and jumps in [6]. The DGFEM for (1) with symmetric interior penalty reads as: Seek uh ∈ Vh such that Ah(uh, v) = (f , v) ∀v ∈ Vh, (3) where