A-posteriori error analysis of hp-version discontinuous Galerkin finite element methods for second-order quasilinear elliptic problems

Report no. NA-0/1 A-posteriori error analysis of hp-version discontinuous Galerkin finite element methods for second-order quasilinear elliptic problems Paul Houston 1, Endre Süli, and Thomas P. Wihler We develop the a-posteriori error analysis of hp-version interior-penalty discontinuous Galerkin finite element methods for a class of second-order quasilinear elliptic partial differential equations. Computable upper and lower bounds on the error are derived in terms of a natural (mesh-dependent) energy norm. The bounds are explicit in the local mesh size and the local degree of the approximating polynomial. The performance of the proposed estimators within an automatic hp-adaptive refinement procedure is studied through numerical experiments. Key words and phrases: hp-adaptivity, a-posteriori error estimation, discontinuous Galerkin methods, quasilinear elliptic PDEs. Oxford University Computing Laboratory Numerical Analysis Group Wolfson Building Parks Road Oxford, England OX1 QD September, 00 1 School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 RD, UK. Supported by the EPSRC, Grant GR/R71. Paul.Houston@ nottingham.ac.uk Oxford University Computing Laboratory, Numerical Analysis Group, Wolfson Building, Parks Road, Oxford OX1 QD, UK. Endre.Suli@comlab.ox.ac.uk. Department of Mathematics and Statistics, McGill University, 80 Sherbrooke W., Montreal, QC HA K, Canada. wihler@ math.mcgill.ca.

1 Introduction In this article, we consider the a-posteriori error analysis, in a natural mesh-dependent energy norm, for a class of interior-penalty hp-version discontinuous Galerkin finite element methods for the numerical solution of the following quasilinear elliptic boundary value problem: (µ(x, u ) u) = f in Ω, (1.1) u = 0 on Γ. (1.) Here, Ω is a bounded polygonal domain in R with boundary Γ, and f L (Ω). Additionally, we assume that the nonlinearity µ satisfies the following assumptions: (A1) µ C(Ω [0, )); (A) there exist positive constants m µ and M µ such that m µ (t s) µ(x, t)t µ(x, s)s M µ (t s), t s 0, x Ω. (1.) We remark that, if µ satisfies (1.), there exist constants C 1 and C, C 1 C > 0, such that for all vectors v,w R, and all x Ω, µ(x, v )v µ(x, w )w C 1 v w, (1.) C v w (µ(x, v )v µ(x, w )w) (v w); (1.) see [, Lemma.1]. Nonlinearities of this kind appear in numerous problems in continuum mechanics. In particular, they arise in mathematical models for non-newtonian fluids, such as the following generalised power-law model: given f L (Ω), find (u, p) H 1 (Ω) L (Ω)/R such that {µ(x, e(u) )e(u)} + p = f in Ω, divu = 0 in Ω, u = 0 on Γ, where u = (u 1, u ) T is the velocity vector, p is the pressure, f = (f 1, f ) T is the applied force, e(u) is the symmetric strain tensor defined by e ij (u) = 1 ( ui + u ) j, i, j = 1,, x j x i and e(u) is the Frobenius norm of e(u). For the sake of notational simplicity we shall suppress the dependence of µ on x and write µ(t) instead of µ(x, t). Indeed, in many physical applications µ is in fact independent of x; for example, in the Carreau law for a non-newtonian fluid, we have µ(t) = µ + (µ 0 µ )(1 + λt ) r, where λ > 0, 1 < r and 0 < µ < µ 0.

In recent years there has been considerable interest in discontinuous Galerkin finite element methods for the numerical solution of a wide range of partial differential equations. We shall not attempt to give an extensive survey of this area of research: the reader is referred to [11] for a detailed review. Discontinuous Galerkin Finite Element Methods (DGFEMs) were introduced in the early 1970s for the numerical solution of first-order hyperbolic problems (see [1,1,1,9,,1]). Simultaneously, but quite independently, they were proposed as nonstandard schemes for the approximation of second-order elliptic equations [1, 8, ]. The recent upsurge of interest in this class of methods has been stimulated by the computational convenience of DGFEMs due to a high degree of locality, the need to approximate advection-dominated diffusion problems without excessive numerical stabilisation, the necessity to accommodate high-order hp- and spectral element discretisations for first-order hyperbolic equations and advection-diffusion problems [17, 1], and the desire to handle nonlinear hyperbolic problems in a locally conservative manner and without auxiliary numerical stabilisation [9, 1]; see also [8, 10] for the error analysis of the local version of the DGFEM in the elliptic case, as well as [] and [9]. In the recent article [0] a family of interior-penalty hp-dgfems was formulated for the numerical approximation of the scalar quasilinear boundary value problem (1.1) (1.), and a-priori bounds were derived on the error, measured in terms of a mesh-dependent energy norm. These error bounds are optimal with respect to the mesh size h and mildly suboptimal (by p 1/ ) in the polynomial degree p; more precisely, for u C 1 (Ω) H k (Ω), k, it was shown that, for any member of the family of methods considered, the error tends to zero at the rate O(h s 1 /p k / ), where 1 s min{p + 1, k}, as h tends to zero and p tends to infinity. For related work on h-version local DGFEMs for quasilinear PDEs, we refer to the articles [7, 18], for example. Here, we extend this work by considering the a-posteriori error analysis of the interior-penalty hp-dgfems from [0]. In particular, we shall derive computable upper and lower bounds on the error, measured in terms of the underlying DG-energy norm, which are explicit in terms of their dependence on h and p. The upper bound is based on the general techniques developed in the articles [1,,, ]. Indeed, here the proof crucially relies on the approximation of discontinuous finite element functions by conforming ones. Results of this type have been developed independently by a number of authors in the context of the h-version of the DGFEM; see, for example, [19, 0, ]. The extension of this type of result to the hp-version of the DGFEM was recently undertaken in the article []. In contrast to [], for example, here we avoid the need to introduce an auxiliary formulation of the underlying DGFEM through the use of lifting operators, while still only requiring minimal regularity assumptions on the unknown analytical solution. The proof of the lower (efficiency) bounds is based on the techniques presented in []. As in the case of the conforming hp-version finite element methods considered in [], reliability and efficiency of our error bounds cannot be established uniformly with respect to the polynomial degree, since the proof of efficiency relies on employing inverse estimates which are suboptimal in the spectral order. Finally, numerical

experiments highlighting the performance of the proposed estimator within an hpadaptive mesh refinement algorithm will also be presented. The outline of this article is as follows. In Section, we revisit the hp-dgfem introduced in [0], for the numerical approximation of the boundary-value problem (1.1) (1.). In Section, our a-posteriori error bounds are presented and discussed; both upper and lower energy norm bounds will be derived. In Section, we present a series of numerical experiments to illustrate the performance of the proposed error estimator within an automatic hp-mesh refinement algorithm. Finally, in Section we summarise the main results of this article and draw some conclusions. Throughout the paper, we use the following standard function spaces. For a bounded Lipschitz domain D R d, d 1, we write H t (D) to denote the usual (real) Sobolev space of order t 0 with norm t,d. In the case t = 0, we set L (D) = H 0 (D). We define H 1 0(D) to be the subspace of functions in H 1 (D) with zero trace on D. For a function space X(D), we write X(D) d to denote the space of d-component vector fields whose components belong to X(D); this space is equipped with the usual product-norm which, for simplicity, is denoted in the same way as the norm in X(D). hp-version discontinuous Galerkin FEM Let T h be a subdivision of Ω into disjoint open element domains such that Ω =. We assume that the family of subdivisions {T h } h>0 is shape-regular (see, for example, [, pp. 1, 11, and Remark., p. 11]) and each T h is an affine image of a fixed master element ; i.e., for each T h there exists an affine mapping F : such that = F ( ), where is either the open unit triangle {(x, y) : 1 < x < 1, 1 < y < x}) or the open unit square ( 1, 1) in R. By h we denote the element diameter of T h, h = max Th h, and n signifies the unit outward normal vector to. We allow the meshes T h to be 1-irregular, i.e., each edge of any one element T h contains at most one hanging node (which, for simplicity, we assume to be the midpoint of the corresponding edge). Here, we suppose that T h is regularly reducible (cf. [0, Section 7.1]), i.e., there exists a shaperegular conforming (regular) mesh T h (consisting of triangles and parallelograms) such that the closure of each element in T h is a union of closures of elements of T h, and that there exists a constant C > 0, independent of the element sizes, such that for any two elements T h and T h with we have h /h C. Note that these assumptions imply that the family {T h } h>0 is of bounded local variation, i.e., there exists a constant ρ 1 1, independent of the element sizes, such that ρ 1 1 h /h ρ 1, (.1) for any pair of elements, T h which share a common edge e =. For a nonnegative integer k, we denote by P k ( ) the set of polynomials of total degree k on. When is the unit square, we also consider Q k ( ), the set of all

tensor-product polynomials on of degree k in each co-ordinate direction. To each T h we assign a polynomial degree p (local approximation order). We store the h, p and F in the vectors h = {h : T h }, p = {p : T h } and F = {F : T h }, respectively, and consider the finite element space S p (Ω, T h,f) = {v L (Ω) : v F R p ( ) T h }, where R is either P or Q. We shall suppose that the polynomial degree vector p, with p 1 for each T, has bounded local variation, i.e., there exists a constant ρ 1 independent of h and p, such that, for any pair of neighbouring elements, T h, ρ 1 p /p ρ. (.) Let us consider the set E of all one-dimensional open edges, or, simply, edges, of all elements T h. Further, we denote by E int the set of all edges e in E that are contained in Ω (interior edges). Additionally, let Γ int = {x Ω : x e for some e E int }, and introduce E B to be the set of boundary edges consisting of all e E that are contained in Ω. Moreover, let = Γ int Γ. Suppose that e is an edge of an element T h. Then, by h e, we denote the length of e. Due to our assumptions on the subdivision T h we have that, if e, then h e is commensurate with h, the diameter of. Given that e E int, there exist indices i and j such that i > j and i, j T h share the edge e; we define the (element-numbering-dependent) jump of an (elementwise smooth) function v across e and the mean-value of v on e by [v] e = v i e v j e and v e = 1 ( ) v i e + v j e, respectively. For a boundary edge e Γ, and thereby e Γ for some T, we define [v] e = v e = v e. When there is no danger of confusion, the subscript e will be suppressed. Additionally, we associate with each edge e Γ int the unit normal vector ν which points from i to j (i > j); if e Γ, then ν is defined as the outward unit normal vector on Γ. With these notations and a parameter θ [ 1, 1], we introduce the semilinear form B(w, v) = µ( h w ) h w h v dx Ω µ( h w ) h w ν [v] ds + θ µ(h 1 [w] ) h v ν [w] ds Γ int,b + σ [w] [v] ds, (.)

and the linear functional l(v) = Ω fv dx. (.) Here, h denotes the element-wise gradient operator defined, for v H 1 (Ω, T h ), by ( h v) = (v ). For an edge e E, the discontinuity penalisation parameter σ, featuring in B(, ) above, is defined by σ e = σ e = γ p e h e, (.) where γ 1 is a (sufficiently large) constant, cf. Theorem below. The hp-dgfem approximation of problem (1.1) (1.) reads as follows: find u DG S p (Ω, T,F) such that B(u DG, v) = l(v) v S p (Ω, T,F). (.) Remark 1 In the case of an inhomogeneous Dirichlet boundary condition, u = g on Γ, the third term in the semilinear form B DG (, ) must be replaced by θ µ(h 1 [w] ) h v ν [w] ds + θ µ(h 1 w g ) h v n (w g) ds, Γ int Γ while the linear functional l( ) defined in (.) must be substituted by l(v) = fv dx + σgv ds; we refer to [0] for further details. Ω The existence and uniqueness of the DG solution u DG satisfying (.) is guaranteed by the following result proved in [0, Theorem.]. Theorem Suppose that γ in (.) is chosen sufficiently large. Then, there exists a unique element u DG in S p (Ω, T,F) such that (.) holds. We conclude this section by equipping the DG space S p (Ω, T,F) with the DG energy norm DG defined by ( ) 1/ v DG = v dx + σ [v] ds T induced by the energy inner product (v, w) DG = v w dx + σ [v] [w] ds. T The a-priori error analysis of the discontinuous Galerkin finite element method (.) has been developed in [0]; here, we shall be concerned with its a-posteriori error analysis. Γ

7 hp-version a-posteriori error analysis Under the structural hypotheses (1.) (1.) on the coefficient µ, the existence of a unique solution u H 1 0 (Ω) to (1.1) (1.) follows from the following result from the theory of monotone operators (see [7], Theorem..), with H = H 1 0(Ω), Λ = C 1 and λ = C. Henceforth, we shall therefore assume that u H 1 0 (Ω). Proposition Let H be a real Hilbert space with inner product (, ) H and norm H, and let T be an operator from H into itself. Suppose that T is Lipschitzcontinuous on H, i.e. there exists Λ > 0 such that T(w 1 ) T(w ) H Λ w 1 w H w 1, w H, and strongly monotone on H, i.e. there exists λ > 0 such that (T(w 1 ) T(w ), w 1 w ) H λ w 1 w H w 1, w H. Then, T is a bijection of H onto itself, and the inverse T 1 of T is Lipschitzcontinuous on H:.1 Upper bound T 1 f T 1 g H (Λ/λ) f g H f, g H. In this section we will formulate the main result of this paper, Theorem. To this end, we first define, for an element T h and an edge e E int, the data-oscillation terms and O (1) = h p (I Π T h ) (f + (µ( u DG ) u DG )) 0,, (.1) O () e = h e p 1 e (I Π E ) e ([µ( u DG u DG ) ν]) 0,e, (.) which depend on the right-hand side f in (1.1) and the numerical solution u DG from (.). Here, I is a generic identity operator and Π Th denotes the element-wise L -projector onto the space S p 1 (Ω, T h,f), where p 1 = {p 1} Th. Furthermore, Π E e is defined as the L -projector onto P pe 1(e); here, p e = max{p, p }, with, T h, e = (we note that, due to our assumptions on the polynomial degree vector p, the quantities p e, p and p are all commensurate with one another). Theorem Let the analytical solution u of (1.1) (1.) belong to H 1 0(Ω). Furthermore, let u DG S p (Ω, T h,f) be its discontinuous Galerkin approximation, i.e. the solution of (.). Then, the following hp-version a-posteriori error bound holds: u u DG DG C ( η + O(f, u DG), (.)

8 where the local error indicators η, T h, are defined by and η = h p Π T h (f + (µ( u DG ) u DG )) 0, + h p 1 Π E([µ( u DG ) u DG ν]) 0, \Γ + γ h 1 p [u DG ] 0,, (.) O(f, u DG ) = O (1) + O e (). e E int Here, C > 0 is a constant that is independent of h, the polynomial degree vector p, and the parameters γ and θ, and only depends on the shape-regularity of the mesh and on the constants ρ 1 and ρ from (.1) and (.), respectively. Remark We observe a slight suboptimality with respect to the polynomial degree in the last term of the local error estimator η in (.). This results from the fact that, due to the possible presence of hanging nodes in T h, a nonconforming interpolant is used in the proof of the above Theorem ; cf. Section.1.. Indeed, for conforming (regular) meshes, i.e. meshes without any hanging nodes, a conforming (hp-version) Clément interpolant, as constructed in [], can be employed; this then results in an a posteriori error bound of the form (.), with the term γ h 1 p [u DG ] 0, in (.) replaced by the improved expression γh 1 p [u DG ] 0, ; cf. []. Remark In order to incorporate the inhomogeneous boundary condition u = g on Γ, the error indicators η are simply adjusted by modifying the jump indicators σ 1 [u DG ] 0, on Γ, with the inclusion of an additional data-oscillation term; see [] for details..1.1 DG decompositions The hp-version a-posteriori error analysis for the DGFEM (.) will be based on an approach similar to the one discussed in [] (see also [1,,,], for related work). In contrast with the analysis presented in [] though, here we shall also admit 1-irregular meshes containing hanging nodes. To this end, consider a given subdivision T h of Ω that is regularly reducible, i.e., T h can be refined to a shaperegular conforming mesh T h as described in Section. Furthermore, denote by S p (Ω, T h, F) the corresponding DG space, with a suitable affine element mapping vector F and a polynomial degree vector p that is defined by p = p, for any T h with, for some T h. We note that S p (Ω, T h,f) S p (Ω, T h, F), and that, due to our assumptions in Section (specifically, the commensurability of the local element sizes and of the local polynomial degrees in T h and T h, due to our bounded local variation assumptions), the DG energy norms DG and DG corresponding to the DG spaces S p (Ω, T h,f) and S p (Ω, T h, F), respectively, are equivalent on S p (Ω, T h,f); in particular, there exist positive constants N 1, N,

9 independent of h and p, such that N 1 σ [v] Γ ds σ [v] ds N σ [v] ds v S p (Ω, T h,f). int,b (.) Here, denotes the union of all element edges of T h, and σ is the discontinuity penalisation parameter on S p (Ω, T h, F) which is defined analogously as for S p (Ω, T h,f) in (.); note that, for v S p (Ω, T h,f), the jump [v] vanishes on \. An important step in our analysis is the decomposition of the space S p (Ω, T h, F) into two orthogonal subspaces: a conforming part [S p (Ω, T h, F)] = S p (Ω, T h, F) H 1 0 (Ω), and a nonconforming part [S p (Ω, T h, F)] defined as the orthogonal complement of [S p (Ω, T h, F)] in S p (Ω, T h, F) with respect to the DG energy inner product (, ) DG (inducing the DG energy norm DG ), i.e., S p (Ω, T h, F) = [S p (Ω, T h, F)] DG [S p (Ω, T h, F)]. Based on this setting, the DG-solution u DG obtained by (.) may be split accordingly, u DG = u DG + u DG, (.) where u DG [S p (Ω, T h, F)] and u DG [S p (Ω, T h, F)]. Furthermore, we define the error of the hp-dgfem by e DG = u u DG, (.7) and let.1. Auxiliary results e DG = u u DG H1 0 (Ω). (.8) For the proof of the above Theorem, we shall require the following auxiliary results. Proposition 7 Under the assumptions in Section on the (regularly reduced) subdivision T h, the following norm-equivalence holds over the space [S p (Ω, T h, F)] : C 1 v DG σ [v] ds C v DG v [S p (Ω, T h, F)], (.9) where the constants C 1, C > 0 depend only on the shape-regularity of T h and on the constants ρ 1 and ρ in (.1) and (.), respectively. Proof See [, Proposition.]. defined by (.) and (.8), respectively, the fol- Corollary 7A With u DG and e DG lowing bounds hold: u DG DG D 1 ( σ [u DG ] ds, e DG DG D e DG DG,

10 where the constants D 1, D > 0 are independent of γ, h and p, and only depend on the shape-regularity of T h and the constants ρ 1 and ρ in (.1) and (.), respectively. Proof In order to prove the first of the above bounds, we recall that u DG H1 0 (Ω). This implies that [u DG ] = 0 on, and hence, [u DG] = [u DG ] + [u DG] = [u DG + u DG] = [u DG ]. Then, due to Proposition 7, we obtain, u DG DG C σ [u DG] ds = C σ [u DG ] ds. (.10) Furthermore, since u DG S p (Ω, T h,f), and because of (.), we conclude that u DG DG C σ [u DG ] ds. For the second bound, we use the triangle inequality, the bound (.10), and the fact that, since the analytical solution u of (1.1) (1.) and e DG belong to H1 0 (Ω), we have [u] = [e DG ] = 0 and [e DG ] = [u] [u DG ] = [u DG ] (.11) on (and thereby also on ). Thus, e DG DG = e DG DG e DG DG + u DG DG e DG DG + C e DG DG + C In a similar way, we obtain e DG DG = Th ( Γint,B σ [e DG ] ds C e DG DG. ( Γint,B e DG 0, + Γint,B σ [e DG ] ds = e DG 0, + σ [u DG ] ds (.1) σ [u DG ] ds. Moreover, observing that u DG S p (Ω, T h,f), and applying (.), leads to e DG DG e DG 0, + C σ [u DG ] ds T h = e DG 0, + C σ [e DG ] ds C e DG DG, T h which, referring to (.1), yields the second bound. Next, we state the following approximation property.

11 Lemma 8 For any ϕ H 1 0(Ω), there exists a function ϕ hp S p (Ω, T h,f) such that h p ϕ ϕ hp 0, + (ϕ ϕ hp) 0, + h 1 p ϕ ϕ hp 0, C I ϕ 0,, (.1) for any T h, with an interpolation constant C I > 0, which is independent of h and p, and only depends on the shape-regularity of the mesh and the constants ρ 1 and ρ in (.1) and (.), respectively. Proof We first consider the proof of the upper bounds on the L ()-norms of ϕ ϕ hp and (ϕ ϕ hp ). In this case, on quadrilateral elements, the above approximation property follows from the tensorisation of the corresponding one-dimensional approximation results for an H 1 -projector; see [], for details. For triangular elements, we employ a reflection technique. More precisely, writing to denote the canonical triangle with vertices ( 1, 1), (1, 1), and ( 1,1), we define to be triangle with vertices (1, 1), (1,1), and ( 1,1) obtained by reflecting about its longest edge. Analogously, given v H 1 ( ), we write v H 1 ( ) to denote the reflection of v in the line ξ = ξ 1, where (ξ 1,ξ ) denotes the local coordinate system for the reference element. With this notation we define the function w H 1 (Ŝ) by w = v and w = v, where Ŝ is the unit square ( 1,1). Due to symmetry, we deduce that there exists a positive constant C, such that v 0, ŵ 0, S C v 0, and v 0, ŵ 0, S C v 0,. Thereby, the approximation properties on the reference element now follow from the corresponding results on the unit square Ŝ; the proof is then completed by employing a standard scaling argument. The upper bound on the approximation error measured in terms of the L ( )-norm now follows from the above results, together with the trace inequality v 0, C ( v 0, v 0, + h 1 v 0,), where v H 1 () and C is a positive constant which depends only on the shape-regularity of the element..1. Proof of Theorem We commence the proof of our main theorem by applying (1.). This yields ( ) C e DG DG = C u u DG dx + σ [e DG ] ds T h = C u u DG dx + σ [e DG ] ds Th (µ( u ) u µ( u DG ) u DG ) e DG dx + C σ [e DG ] Γ ds int,b Th T 1 + T + T, (.1)

1 where T 1 = Th T = Th (µ( u ) u µ( u DG ) u DG ) e DG dx, T = C σ [e DG ] ds, (µ( u ) u µ( u DG ) u DG ) u DG dx, and e DG H1 0 (Ω) and u DG [S p (Ω, T h, F)] are defined as in (.) and (.8), respectively. We will now analyse the three terms T 1, T and T separately. Term T 1. We first note that T 1 = (µ( u ) u µ( u DG ) u DG ) e DG dx. Then, using integration by parts, we obtain T 1 = (µ( u ) u)e DG dx µ( u DG ) u DG e DG dx = fe DG dx µ( u DG ) u DG e DG dx. We now let ϕ hp S p (Ω, T h,f) be the element-wise projection of e DG satisfying Lemma 8. Then, by the definition of the hp-dgfem (.), it follows that T 1 = f(e DG ϕ hp) dx µ( u DG ) u DG (e DG ϕ hp) dx µ( h u DG ) h u DG ν [ϕ hp ] ds + θ µ(h 1 [u DG ] ) h ϕ hp ν [u DG ] ds Γ int,b + σ [u DG ] [ϕ hp ] ds.

1 Hence, integrating the second term on the right-hand side of the above equality by parts, leads to T 1 = (f + (µ( u DG ) u DG ))(e DG ϕ hp) dx (µ( u DG ) u DG n )(e DG ϕ hp) ds µ( h u DG ) h u DG ν [ϕ hp ] ds + θ µ(h 1 [u DG ] ) h ϕ hp ν [u DG ] ds Γ int,b + σ [u DG ] [ϕ hp ] ds. Using the fact that [e DG ] = 0 on, since e DG H1 0 (Ω), and a few elementary calculations, we have that (µ( u DG ) u DG n )(e DG ϕ hp) ds = [µ( h u DG ) h u DG ν] e DG ϕ hp ds + Γ int Therefore, T 1 = and thus, T 1 (f + (µ( u DG ) u DG )) (e DG ϕ hp) dx µ( h u DG ) h u DG ν [ϕ hp ] ds. [µ( h u DG ) h u DG ν] e DG ϕ hp ds Γ int + θ µ(h 1 [u DG ] ) h ϕ hp ν [u DG ] ds + σ [u DG ] [ϕ hp ] ds, + f + (µ( u DG ) u DG ) e DG ϕ hp dx \Γ [µ( u DG ) u DG ν] e DG ϕ hp ds + θ h µ(h 1 [u DG ] )(h 1 [u DG ] ) h ϕ hp ν ds + f + (µ( u DG ) u DG ) 0, e DG ϕ hp 0, σ [u DG ] [ϕ hp ] ds + C [µ( u DG ) u DG ν] 0, \Γ e DG ϕ hp 0, + M µ θ [u DG ] h ϕ hp ds + σ [u DG ] [ϕ hp ] ds,

1 where we have applied (1.) (with s = 0 and t = h 1 [u DG ] ) to bound the secondlast of the above terms. Moreover, proceeding as in the proof of [0, Lemma.] (cf., also [, Lemma.]) and recalling that γ 1, we obtain [u DG ] h ϕ hp ds ( C σ [u DG ] ds ( σ [u DG ] ds ( e E e ( h p ϕ hp ds ϕ hp 0,. Furthermore, by (.1), we have ϕ hp 0, C (e DG ϕ hp) 0, + C e DG 0, C e DG 0,, and hence, [u DG ] h ϕ hp ds C ( σ [u DG ] ds ( e DG 0, Moreover, using again the fact that [e DG ] = 0 on, and recalling (.1) (.), implies σ [u DG ] [ϕ hp ] ds = σ [u DG ] [e DG ϕ hp] ds ( ( C σ p [u DG ] ds σ p 1 e DG ϕ hp ds T h ( ( Cγ 1 σ p [u DG ] ds h 1 p e DG ϕ hp 0,. Thus, collecting the terms leads to T 1 h p 1 f + (µ( u DG ) u DG ) 0, h 1 p e DG ϕ hp 0,. + C h 1 p 1 [µ( u DG ) u DG ν] 0, \Γ h 1 p 1 e DG ϕ hp 0, + C θ + Cγ 1 ( ( σ [u DG ] ds σ p [u DG ] ds ( e DG 0, ( h 1 p e DG ϕ hp 0,.

1 Furthermore, applying again the approximation property (.1), using that γ 1 θ 0, and incorporating (.), results in T 1 C ( h p f + (µ( u DG ) u DG ) 0, ( + C + C θ + Cγ 1 + C + C Therefore, ( h p 1 [µ( u DG ) u DG ν] 0, \Γ ( ( ( σ [u DG ] ds σ p [u DG ] ds ( e DG 0, ( h p f + (µ( u DG ) u DG ) 0, ( ( h p 1 [µ( u DG ) u DG ν] 0, \Γ γ ( h 1 p [u DG ] 0, which, by Corollary 7A, yields T 1 C T 1 C ( η ( η Here, for T h, the term η is defined by h p e DG ϕ hp 0, ( h 1 p e DG ϕ hp 0, ( ( e DG 0, e DG DG, e DG DG. h 1 p e DG ϕ hp 0, e DG 0, e DG 0,. η = h p f + (µ( u DG ) u DG ) 0, + h p 1 [µ( u DG ) u DG ν] 0, \Γ + γ h 1 p [u DG ] 0,. Noticing that η C η + O(1) + e E int e O () e,

1 we obtain T 1 C η + O (1) + e E int e O () e 1 e DG DG. (.1) Term T. In order to bound T we recall (1.). This yields T µ( u ) u µ( u DG ) u DG u DG dx Th C 1 e DG u DG dx C 1 e DG 0, u Th DG 0, Th 1 1 C 1 e DG 0, u Th DG 0,. Th Hence, we have T C 1 e DG DG u DG DG, which, upon applying Corollary 7A, gives T C e DG DG ( σ [u DG ] ds C e DG DG ( γ h 1 p [u DG ] 0,, and thus, since γ 1, T C e DG DG ( η. (.1) Term T. A bound for T is found by recalling (.11). This gives T C σ [e DG ] [u DG ] ds C C e DG DG ( ( γ h 1 p [u DG ] 0, σ [e DG ] ds. ( σ [u DG ] ds Thereby, we obtain T C e DG DG ( η. (.17)

17 Finally, combining the bounds (.1) and (.1) (.17) leads to e DG DG C ( η + O(f, u DG) e DG DG. Dividing both sides of the above inequality by e DG DG completes the proof of Theorem.. Local lower bounds In this section we derive local lower bounds on the error measured in terms of the DG energy norm DG. As in the case of conforming hp-version finite element methods, estimators which are both optimally reliable and efficient in the polynomial degree are not currently available in the literature, cf. [], for example. The key technical reason for this is that the proofs of the lower bounds exploit the use of inverse estimates which are suboptimal in the polynomial degree. To minimise the deterioration of the efficiency bounds with respect to the polynomial degree, weighted versions of the local a-posteriori error indicators η may be employed. This idea was first used in the context of conforming finite element methods in []; subsequent extensions to DGFEMs have been undertaken in the article [], for example. For simplicity of exposition, we only present lower bounds for our (unweighted) a posteriori error indicators η ; extensions to weighted versions of η follow analogously, cf. []. We begin by quoting the following theorem under the assumption that the computational mesh T h is conforming (regular). The extension of these bounds to nonconforming (irregular) meshes which are regularly reducible follows analogously; cf. Remark 10 below. Theorem 9 Let, T h be any two neighbouring elements, e = E int, and ω e = ( ). Then, for all δ > 0, the following local hp-version a-posteriori lower bounds on the error e DG from (.7) hold: a) b) ( ) Π Th (f + (µ( u DG ) u DG )) 0, Ch 1 p e DG 0, + p δ 1 O (1) ; Π E e (([µ( u DG ) u DG ] ν) e ) 0,e Ch 1 p δ+ e DG 0,ωe + p δ 1 τ {, } O τ (1) + p 1 O () e ; c) [u DG ] 0,e Cγ 1 h 1 p 1 σ 1 [edg ] 0,e.

18 Here, the generic constant C > 0 depends on δ, but is independent of h and p. Proof We proceed similarly as in []; see also []. To this end, we first introduce suitable cut-off functions as follows: on the reference element, we define a weightfunction Φ (x) = min y x y. Then, for T h, we let Φ = c Φ F 1, where the factor c is chosen so that (Φ 1)dx = 0. Furthermore, on the reference interval ι = ( 1,1), we define the weight-function Φ ι(x) = 1 x. Then, for an interior edge e E int, we let Φ e = c e Φ ι Fe 1, where F e is the affine mapping from ι to e, and c e is chosen so that e (Φ e 1) ds = 0. Proof of a): Let T h and define v = Φ α Π Th (f + (µ( u DG ) u DG )), where α ( 1,1]. Then, using (1.1), and integrating by parts, yields Φ α v 0, = v Π Th (f + (µ( u DG ) u DG ))dx = v (µ( u DG ) u DG µ( u ) u)dx + v (Π Th I)(f + (µ( u DG ) u DG ))dx = v (µ( u DG ) u DG µ( u ) u)dx + v (Π Th I)(f + (µ( u DG ) u DG ))dx v µ( u DG ) u DG µ( u ) u dx + v (Π Th I)(f + (µ( u DG ) u DG )) dx. Recalling (1.), this can be transformed into Φ α v 0, C v e DG dx + v (Π Th I)(f + (µ( u DG ) u DG )) dx C v 0, e DG 0, + Φ α v 0, Φ α (I Π Th )(f + (µ( u DG ) u DG )) 0, C v 0, e DG 0, + h 1 p Φ α v 0, O (1). From the proof of [, Lemma.], we have thereby, v 0, Ch 1 p α Φ α v 0, ; ( ) Φ α v 0, Ch 1 p Φ α v 0, p 1 α e DG 0, + O (1). Dividing both sides of the above inequality by Φ α v 0, and observing that (by applying the inverse inequality from [, Theorem.]) Π Th (f + (µ( u DG ) u DG )) 0, Cp α Φ α Π Th (f + (µ( u DG ) u DG )) 0, = Cp α Φ α v 0,,

19 leads to ( ) Π Th (f + (µ( u DG ) u DG )) 0, Ch 1 p 1+α p 1 α e DG 0, + O (1). (.18) Choosing δ = α 1, completes the proof of a). Proof of b): Let q e = Φ α e Π E e (([µ( u DG ) u DG ] ν) e ), where again α ( 1,1]. Then, referring to [, Lemma. with ε = p ], there exists χ e H 1 0 (ω e) such that χ e e = q e and χ e 0,ωe Ch 1 p 1 Φ α e q e 0,e, χ e 0,ωe Ch 1 p Φ α e q e 0,e. (.19) Noting that (µ( u ) u) = f L (Ω), we conclude that [µ( u ) u] ν = 0 on e. Hence, integrating by parts and assuming (without loss of generality) that the normal vector ν points from to, leads to Φ α e q e 0,e = Π E ([µ( u DG ) u DG ] ν)χ e ds e = ([µ( u DG ) u DG µ( u ) u] ν)χ e ds + (Π E I)([µ( u DG ) u DG ] ν)χ e ds e e = ((µ( u DG ) u DG µ( u ) u) n )χ e ds + ((µ( u DG ) u DG µ( u ) u) n ) χ e ds + (Π E I)([µ( u DG ) u DG ] ν)χ e ds e = (µ( u DG ) u DG µ( u ) u) χ e dx + (f + (µ( u DG ) u DG ))χ e dx ω e ω e + (Π E I)([µ( u DG ) u DG ] ν)χ e ds e R 1 + R + R. (.0) Employing (1.) and (.19), R 1 can be bounded as follows: R 1 C e DG χ e dx C e DG 0,ωe χ e 0,ωe Ch 1 p e DG 0,ωe Φ α e q e 0,e. ω e (.1)

0 In order to obtain a bound for R, we use (.18) and the definition of O (1) from (.1); thereby, R = Π Th (f + (µ( u DG ) u DG ))χ e dx ω e (Π Th I)(f + (µ( u DG ) u DG ))χ e dx ω e Recalling (.19), this gives Π Th (f + (µ( u DG ) u DG )) 0,ωe χ e 0,ωe + (Π Th I)(f + (µ( u DG ) u DG )) 0,ωe χ e 0,ωe Ch 1 p 1+α p 1 α e DG 0,ωe + O τ (1) χ e 0,ωe. R Ch 1 p α p 1 α e DG 0,ωe + A bound on R is based on the definition of O e () on e: τ {, } τ {, } O (1) τ Φ α e q e 0,e. (.) from (.) and on the fact that χ e = q e R Φ α e (Π E I)([µ( u DG ) u DG ] ν) 0,e Φ α e χ e 0,e Ch 1 p 1 O e () Φ α e q e 0,e. (.) Combining (.0) (.), gives Φ α e q e 0,e Ch 1 p e DG 0,ωe + p α 1 τ {, } O τ (1) + p 1 O () e Φ α e q e 0,e. As in the proof of a), we divide the above inequality by Φ α e q e 0,e, and use the fact that Φ α e q e = Φ α e Π E e (([µ( u DG ) u DG ] ν) e ). Then, applying the inverse inequality from [, Lemma.] (see also [, ]), we get Π E ([µ( u DG ) u DG ] ν) 0,e Cp α Φ α e Π E ([µ( u DG ) u DG ] ν) 0,e = Cp α Φ α e q e 0,e. Thereby, Π E ([µ( u DG ) u DG ] ν) 0,e Ch 1 p 1+α e DG 0,ωe + p α 1 τ {, } Again, selecting δ = α 1 leads to estimate b). Proof of c): This follows from (.1), (.) and (.11): O τ (1) + p 1 [u DG ] 0,e = [e DG ] 0,e Cγ 1 h 1 p 1 σ 1 [edg ] 0,e. O () e. That completes the proof of the lower bounds.

1 Remark 10 For the case when the mesh T h is 1-irregular (but assumed to be regularly reducible to a conforming mesh T h, cf. Section ), analogous bounds to the ones derived in Theorem 9 still hold. Indeed, bounds a) and c) follow directly; for the proof of b), employing the argument outlined in the proof of Theorem 9, we deduce that Π E e (([µ( u DG ) u DG ] ν) e ) 0,e Ch 1 pδ+ e DG 0, ω e + p δ 1 τ {, } O τ (1) + p 1 O () e, (.) where ω e is defined so that the closure of ω e is the union of the closure of the two elements, T h which share the common edge e. The right-hand side of (.) may now be bounded from above by an similar expression involving quantities measured over the (nonmatching) elements and which share the edge e; by this we mean that, in the estimate (.), the element size h and polynomial degree p are commensurate with h and p, respectively, and the error term e DG 0, ω e is bounded from above by e DG 0,ωe. We note that the data oscillation terms O τ (1) appearing in (.) are, however, still measured over the elements, T h since they are in general not bounded by the corresponding oscillations on, T h. Numerical experiments In this section we present a series of numerical examples to demonstrate the practical performance of the proposed a-posteriori error estimator derived in Theorem within an automatic hp-adaptive refinement procedure which is based on 1-irregular quadrilateral elements. In each of the examples shown in this section the DG solution u DG defined by (.) is computed with θ = 0, i.e., we employ an incomplete-interiorpenalty-type discontinuous Galerkin method. Analogous results to those presented for θ = 0 are also observed with θ = 1 and θ = 1; for brevity these results have been omitted. Additionally, we set the constant γ appearing in the definition of the interior-penalty parameter σ defined in (.) equal to 10. The resulting system of nonlinear equations is solved by employing a damped Newton method; within each inner (linear) iteration, we exploit a (left-) preconditioned GMRES algorithm using a block symmetric Gauss Seidel preconditioner. The hp-adaptive meshes are constructed by first marking the elements for refinement or derefinement according to the size of the local error indicators η ; this is achieved by employing the fixed fraction strategy, see [7], with refinement and derefinement fractions set to % and 10%, respectively. Once an element T h has been flagged for refinement or derefinement, a decision must be made whether the local mesh size h or the local degree p of the approximating polynomial should be adjusted accordingly. The choice to perform either h-refinement/derefinement or p-refinement/derefinement is based on estimating the local smoothness of the

(unknown) analytical solution. To this end, we employ the hp-adaptive strategy developed in [8], where the local regularity of the analytical solution is estimated from truncated local Legendre expansions of the computed numerical solution; see, also, [1,]. Here, the emphasis will be on investigating the asymptotic sharpness of the proposed a-posteriori error bound on a sequence of nonuniform hp-adaptively refined 1-irregular meshes. To this end, we shall compare the estimator derived in Theorem, which is slightly suboptimal (by a factor of p 1/ ) in the spectral order p, with the corresponding optimal one (cf. Remark ); we note that the derivation of the latter precludes the use of hanging nodes. Indeed, here we shall show that despite the loss of optimality in p, the former indicator performs extremely well on hp-refined meshes, in the sense that the effectivity index, which is defined as the ratio of the a-posteriori error bound and the energy norm of the actual error, is roughly constant on all of the meshes employed. Moreover, our numerical experiments indicate that both a-posteriori error indicators give rise to very similar quantitative results. For simplicity, as in [], we set the constant C arising in Theorem equal to one; in general, to ensure the reliability of the error estimator, this constant must be determined numerically for the underlying problem at hand. In all of our experiments, the data-approximation terms in the a-posteriori bound stated in Theorem will be neglected..1 Example 1 In this example, we let Ω be the unit square (0, 1) in R. The nonlinear diffusion coefficient is defined as follows: µ(x, u ) = + 1 1 + u ; further, we select f so that the analytical solution to (1.1) (1.) is given by u(x, y) = x(1 x)y(1 y)(1 y) e s(x 1), where s is a positive constant, cf. [,]; throughout this section we set s = 0. In Figure 1(a) we present a comparison of the actual and estimated energy norm of the error versus the third root of the number of degrees of freedom in the finite element space S p (Ω, T h,f) on a linear-log scale, for the sequence of meshes generated by our hp-adaptive algorithm using the suboptimal indicator stated in Theorem (denoted by p in the figure) and the corresponding optimal one outlined in Remark (denoted by p in the figure). We note that for both indicators meshes employing hanging nodes are employed, despite the fact that the derivation of the latter, hp-optimal, error indicator necessitates the use of conforming (regular) meshes. The third root of the number of degrees of freedom is chosen on the basis of the a-priori error analysis performed in [], for example. Here, we observe that the two error indicators perform in a very similar manner: in each case the error

10 0 10 1 10 10 10 10 Error Estimator (p ) True Error (p ) Error Estimator (p ) True Error (p ) 10 1 0 (Degrees of Freedom) 1 (a) Effectivity Index 0 1 10 p p 0 0 8 10 1 Mesh Number (b) 10 1 10 1 10 10 u udg DG 10 10 u udg DG 10 10 10 h Refinement hp Refinement 10 1 0 0 0 (Degrees of Freedom) 1 (c) 10 h Refinement hp Refinement 10 10 10 Degrees of Freedom (d) Figure 1: Example 1. (a) Comparison of the actual and estimated energy norm of the error with respect to the (third root of the) number of degrees of freedom with hp-adaptive mesh refinement; (b) Effectivity indices; (c) & (d) Comparison of the actual error with h- and hp-adaptive mesh refinement. bound over-estimates the true error by a (reasonably) consistent factor; indeed, from Figure 1(b), we see that the computed effectivity indices oscillate around a value of approximately 1. Additionally, from Figure 1(a) we observe that the convergence lines using hp-refinement are (roughly) straight on a linear-log scale, which indicates that exponential convergence is attained for this smooth problem, as we would expect. In Figures 1(c) & (d), we present a comparison between the actual energy norm of the error employing both h- and hp-mesh refinement; here, the hprefinement is based on employing the error indicator stated in Theorem. In the former case, the DG solution u DG is computed using bilinear elements, i.e., p = 1; here, the adaptive algorithm is again based on employing the fixed fraction strategy, with refinement and derefinement fractions set to % and 10%, respectively. From Figures 1(c) & (d), we clearly observe the superiority of employing a grid adaptation

(a) (b) Figure : Example 1. Finite element mesh after 11 adaptive refinements, with 1198 elements and 18 degrees of freedom: (a) h-mesh alone; (b) hp-mesh.

strategy based on exploiting hp-adaptive refinement: on the final mesh, the energy norm of the error using hp-refinement is over two orders of magnitude smaller than the corresponding quantity computed when h-refinement is employed alone. In Figure we show the mesh generated using the proposed hp-version a-posteriori error indicator stated in Theorem after 11 hp-adaptive refinement steps. For clarity, we show the h-mesh alone, as well as the corresponding polynomial degree distribution on this mesh. Here, we observe that some h-refinement of the mesh has been performed in the vicinity of the base of the exponential hills situated in the left- and the right-hand sides of the domain, where the gradient/curvature of the analytical solution is relativity large. Once the h-mesh has adequately captured the structure of the solution, the hp-adaptive algorithm increased the degree of the approximating polynomial within the interior part of the domain containing these hills.. Example In this section we let Ω be the L-shaped domain ( 1, 1) \[0, 1) ( 1, 0], and select µ(x, u ) = 1 + e u. Then, writing (r, ϕ) to denote the system of polar co-ordinates, we choose f and an appropriate inhomogeneous boundary condition for u so that u = r / sin(ϕ/); cf. [], for example. We note that u is analytic in Ω \ {0}, but u is singular at the origin; indeed, here u H (Ω). Figure (a) shows the history of the actual and estimated energy norm of the error on each of the meshes generated by our hp-adaptive algorithm using both the indicator stated in Theorem (denoted by p in the figure) and the corresponding one outlined in Remark (denoted by p in the figure). As in the previous example, we observe that the two error indicators perform in a very similar manner, though for this nonsmooth example the loss in optimality in the jump indicator in the estimator stated in Theorem does lead to a slight increase in the effectivity indices in comparison with the latter indicator. However, from Figure (b) we observe that asymptotically both a-posteriori bounds over-estimate the true error by a consistent factor. Additionally, from Figure (a) we observe exponential convergence of the energy norm of the error using both estimators with hp-refinement; indeed, on a linear-log scale, the convergence lines are, on average, straight. Figures (c) & (d) highlight the superiority of employing hp-adaptive refinement in comparison with h-refinement: on the final mesh, the energy norm of the error using the hprefinement indicator stated in Theorem is over two orders of magnitude smaller than the corresponding quantity when h-refinement is employed alone, based on using bilinear elements.

10 0 7 10 1 10 10 10 Error Estimator (p ) True Error (p ) Error Estimator (p ) True Error (p ) 8 10 1 1 1 18 (Degrees of Freedom) 1 (a) Effectivity Index 1 p p 0 0 10 1 Mesh Number (b) 10 1 h Refinement hp Refinement 10 1 h Refinement hp Refinement u udg DG 10 10 u udg DG 10 10 10 10 1 0 0 0 (Degrees of Freedom) 1 (c) 10 10 10 10 10 Degrees of Freedom (d) Figure : Example. (a) Comparison of the actual and estimated energy norm of the error with respect to the (third root of the) number of degrees of freedom with hp-adaptive mesh refinement; (b) Effectivity indices; (c) & (d) Comparison of the actual error with h- and hp-adaptive mesh refinement. In Figure we show the mesh generated using the local error indicators η stated in Theorem after 1 hp-adaptive refinement steps. Here, we see that the h-mesh has been refined in the vicinity of the re-entrant corner located at the origin; from the zoom, we see that h-refinement is more pronounced in the direction y = x. In the normal direction, y = x, p-refinement is employed instead, as the solution is deemed to be smooth here. Additionally, we see that the polynomial degrees have been increased away from the re-entrant corner located at the origin, since the underlying analytical solution is smooth in this region.

7 Figure : Example. hp-mesh after 1 adaptive refinements, with 1 elements and 0 degrees of freedom. Concluding remarks In this paper, we derived global upper and local lower residual-based a-posteriori error bounds in the energy norm for the class of interior-penalty hp-dgfems developed in [0] for the numerical approximation of second-order quasilinear elliptic partial differential equations. The analysis is based on employing a suitable DG space decomposition, together with an hp-version projection operator. Numerical experiments presented in this article clearly demonstrate that the proposed a-posteriori estimator converges to zero at the same asymptotic rate as the energy norm of the actual error on sequences of hp-adaptively refined meshes. Future work will be devoted to the extension of our analysis to hp-adaptive discontinuous Galerkin approximations of quasi-newtonian incompressible flow models. References [1] D. Arnold, An interior penalty finite element method with discontinuous elements, SIAM J. Numer. Anal., 19 (198), pp. 7 70.

8 [] D. Arnold, F. Brezzi, B. Cockburn, and L. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal., 9 (001), pp. 179 1779. [] R. Becker, P. Hansbo, and M. Larson, Energy norm a posteriori error estimation for discontinuous Galerkin methods, Comput. Methods Appl. Mech. Engrg., 19 (00), pp. 7 7. [] C. Bernardi and Y. Maday, Spectral methods, in Handbook of Numerical Analysis, P. Ciarlet and J. Lions, eds., vol., North-Holland, Amsterdam, 1997. [] C. Bernardi, R. Owens, and J. Valenciano, An error indicator for mortar element solutions to the Stokes problem, IMA J. Numer. Anal., 1 (001), pp. 87 88. [] D. Braess, Finite Elements. Theory, Fast Solvers, and Applications in Solid Mechanics, Cambridge University Press, 1997. [7] R. Bustinza and G. Gatica, A local discontinuous Galerkin method for nonlinear diffusion problems with mixed boundary conditions, SIAM J. Sci. Comput., (1) (00), pp. 1 177. [8] P. Castillo, B. Cockburn, I. Perugia, and D. Schötzau, An a priori error analysis of the local discontinuous Galerkin method for elliptic problems, SIAM J. Numer. Anal., 8 (000), pp. 17 170. [9] B. Cockburn, S. Hou, and C.-W. Shu, TVB Runge Kutta local projection discontinuous Galerkin finite elements for hyperbolic conservation laws, Math. Comp., (1990), pp. 81. [10] B. Cockburn, G. Kanschat, I. Perugia, and D. Schötzau, Superconvergence of the local discontinuous Galerkin method for elliptic problems on Cartesian grids, SIAM J. Numer. Anal., 9 (001), pp. 8. [11] B. Cockburn, G. Karniadakis, and C.-W. Shu, eds., Discontinuous Galerkin Methods. Theory, Computation and Applications, vol. 11 of Lect. Notes Comput. Sci. Engrg., Springer, 000. [1] B. Cockburn and C.-W. Shu, TVB Runge Kutta local projection discontinuous Galerkin finite element method for scalar conservation laws ii: General framework, Math. Comp., (1989), pp. 11. [1], The Runge Kutta local projection P 1 discontinuous Galerkin method for scalar conservation laws, Modél. Math. Anal. Numér., (1991), pp. 7 1. [1], The local discontinuous Galerkin method for time dependent reaction diffusion systems, SIAM J. Numer. Anal., (1998), pp. 0.

9 [1], The Runge Kutta discontinuous Galerkin method for conservation laws: Multidimensional systems, J. Comput. Phys., 11 (1998), pp. 199. [1] T. Eibner and J. M. Melenk, An adaptive strategy for hp-fem based on testing for analyticity, Tech. Report 1/00, University of Reading, Department of Mathematics, 00. [17] J. Flaherty, R. Loy, M. Shephard, and J. Teresco, Software for parallel adaptive solution of conservation laws by discontinuous Galerkin methods, in Discontinuous Galerkin Methods: Theory, Computation and Applications, Lecture Notes in Computational Science and Engineering, Vol. 11, B. Cockburn, G. Karniadakis, and C.-W. Shu, eds., Springer, 000, pp. 11 1. [18] G. G. M. González and S. Meddahi, A low-order mixed finite element method for a class of quasi-newtonian Stokes flows. Part I: A-priori error analysis, Comput. Methods Appl. Mech. Engrg., 19(9-11) (00), pp. 881 89. [19] P. Houston, I. Perugia, and D. Schötzau, Mixed discontinuous Galerkin approximation of the Maxwell operator, SIAM J. Numer. Anal., (00), pp. 9. [0] P. Houston, J. Robson, and E. Süli, Discontinuous Galerkin finite element approximation of quasilinear elliptic boundary value problems I: The scalar case, IMA J. Numer. Anal., (00), pp. 7 79. [1] P. Houston, D. Schötzau, and T. Wihler, hp-adaptive discontinuous Galerkin finite element methods for the Stokes problem, in Proceedings of the European Congress on Computational Methods in Applied Sciences and Engineering, Volume II, P. Neittaanmäki, T. Rossi, S. Korotov, E. Oñate, J. Périaux, and D. Knörzer, eds., 00. [], Energy norm a posteriori error estimation for mixed discontinuous Galerkin approximations of the Stokes problem, J. Sci. Comput., (1) (00), pp. 7 80. [] P. Houston, D. Schötzau, and T. P. Wihler, Energy norm a posteriori error estimation of hp-adaptive discontinuous Galerkin methods for elliptic problems, Math. Models Methods Appl. Sci., (In press). [] P. Houston, D. Schötzau, and T. P. Wihler, An hp-adaptive mixed discontinuous Galerkin FEM for nearly incompressible linear elasticity, Comput. Methods Appl. Mech. Engrg., (In press). [] P. Houston, C. Schwab, and E. Süli, Stabilized hp-finite element methods for first order hyperbolic problems, SIAM J. Numer. Anal., 7 (000), pp. 118 1.

0 [] P. Houston, B. Senior, and E. Süli, Sobolev regularity estimation for hp adaptive finite element methods, in Numerical Mathematics and Advanced Applications ENUMATH 001, F. Brezzi, A. Buffa, S. Corsaro, and A. Murli, eds., Springer, 00, pp. 1. [7] P. Houston and E. Süli, Adaptive finite element approximation of hyperbolic problems, in Error Estimation and Adaptive Discretization Methods in Computational Fluid Dynamics. Lect. Notes Comput. Sci. Engrg., T. Barth and H. Deconinck, eds., vol., Springer, 00, pp. 9. [8], A note on the design of hp adaptive finite element methods for elliptic partial differential equations, Comput. Methods Appl. Mech. Engrg., 19(-) (00), pp. 9. [9] C. Johnson and J. Pitkäranta, An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation, Math. Comp., (198), pp. 1. [0] O. Karakashian and F. Pascal, A posteriori error estimation for a discontinuous Galerkin approximation of second order elliptic problems, SIAM J. Numer. Anal., 1 (00), pp. 7 99. [1] G. Karniadakis and S. Sherwin, Spectral/hp Finite Element Methods in CFD, Oxford University Press, 1999. [] M. Larson and A. Niklasson, Conservation properties for the continuous and discontinuous Galerkin method, Tech. Report 000-08, Chalmers Finite Element Center, Chalmers University, 000. [] P. LeSaint and P. Raviart, On a finite element method for solving the neutron transport equation, in Mathematical Aspects of Finite Elements in Partial Differential Equations, C. de Boor, ed., Academic Press, New York, 197, pp. 89 1. [] W. B. Liu and J. W. Barrett, Quasi-norm error bounds for the finite element approximation of some degenerate quasilinear elliptic equations and variational inequalities, RAIRO Modél. Math. Anal. Numér., 8 (199), pp. 7 7. [] J. Melenk, hp Interpolation of non-smooth functions, SIAM J. Numer. Anal., (00), pp. 17 1. [] J. Melenk and B. Wohlmuth, On residual-based a posteriori error estimation in hp-fem, Adv. Comp. Math., 1 (001), pp. 11 1. [7] J. Nečas, Introduction to the Theory of Nonlinear Elliptic Equations, John Wiley and Sons, 198.