Discontinuous Galerkin Finite Element Methods for Interface Problems: A Priori and A Posteriori Error Estimations ∗ Zhiqiang Cai † Xiu Ye ‡ Shun Zhang § April 19, 2011 Abstract. Discontinuous Galerkin (DG) finite element methods were studied by many researchers for second-order elliptic partial differential equations, and a priori error esti- mates were established when the solution of the underlying problem is piecewise H 3/2+ǫ smooth with ǫ> 0. However, elliptic interface problems with intersecting interfaces do not possess such a smoothness. In this paper, we establish a quasi-optimal a priori error estimate for interface problems whose solutions are only H 1+α smooth with α ∈ (0, 1) and, hence, fill a theoretical gap of the DG method for elliptic problems with low regularity. The second part of the paper is to design and analyze robust residual- and recovery-based a posteriori error estimators. Theoretically, we show that the residual and recovery estima- tors studied in this paper are robust with respect to the DG norm, i.e., their reliability and efficiency bounds do not depend on the jump, provided that the distribution of coefficients is locally quasi-monotone. 1 Introduction Consider the following interface problem: −∇ · (k(x)∇ u)= f in Ω (1.1) with boundary conditions u = g D on Γ D and n · (k∇ u)= g N on Γ N , (1.2) where f , g D , and g N are given scalar-valued functions; Ω is a bounded polygonal domain in ℜ 2 with boundary ∂ Ω= ¯ Γ D ∪ ¯ Γ N and Γ D ∩ Γ N = ∅; n =(n 1 ,n 2 ) is the outward unit vector normal to the boundary; and diffusion coefficient k(x) is positive and piecewise constant on polygonal subdomains of Ω with possible large jumps across subdomain boundaries (interfaces): k(x)= k i > 0 in Ω i for i =1, ..., n. Here, {Ω i } n i=1 is a partition of the domain Ω with Ω i being an open polygonal domain. Define k min = min 1≤i≤n k i and k max = max 1≤i≤n k i . ∗ This work was supported in part by the National Science Foundation under grant DMS-0810855. † Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, IN 47907-2067. zcai@math.purdue.edu. ‡ Department of Mathematics, University of Arkansas at Little Rock, Little Rock, AK 72204. xxye@ualr.edu. This author was supported in part by National Science Foundation Grant DMS-0813571. § Division of Applied Mathematics, Box F, Brown University, 182 George St., Providence, RI, 02912. Shun_Zhang@brown.edu. 1