MIXED DISCONTINUOUS GALERKIN APPROXIMATION OF THE MAXWELL OPERATOR. SIAM J. Numer. Anal., Vol. 42 (2004), pp

Transcription

1 MIXED DISCONTINUOUS GALERIN APPROXIMATION OF THE MAXWELL OPERATOR PAUL HOUSTON, ILARIA PERUGIA, AND DOMINI SCHÖTZAU SIAM J. Numer. Anal., Vol. 4 (004), pp Abstract. We introduce and analyze a discontinuous Galerkin discretization of te Maxwell operator in mixed form. Here, all te unknowns of te underlying system of partial differential equations are approximated by discontinuous finite element spaces of te same order. For piecewise constant coefficients, te metod is sown to be stable and optimally convergent wit respect to te mes size. Numerical experiments igligting te performance of te proposed metod for problems wit bot smoot and singular analytical solutions are presented. ey words. Discontinuous Galerkin metods, mixed metods, Maxwell operator AMS subject classifications. 65N30. Introduction. Te origins of discontinuous Galerkin (DG) metods can be traced back to te seventies, were tey were proposed for te numerical solution of te neutron transport equation, as well as for te weak enforcement of continuity in Galerkin metods for elliptic and parabolic problems; see [] for a istorical review. In te meantime, tese metods ave undergone quite a remarkable development and are used in a wide range of applications; see te recent survey articles [0,, 3] and te references cited terein. Te main advantages of DG metods lie in teir robustness, conservation properties and great flexibility in te mes-design. Indeed, being based on completely discontinuous finite element spaces, tese metods can easily andle elements of various types and sapes, non-matcing grids and local spaces of different polynomial orders; tus, tey are ideal for p-adaptivity. In recent years, DG metods ave begun to find teir way into computational electromagnetics. Here, we mention te work in [7], were te full Maxwell system is discretized using unstructured spectral elements in space togeter wit a suitable low-storage Runge-utta time stepping sceme; similar spectral DG metods were proposed in []. Te study of DG metods applied to te time-armonic Maxwell equations in electric field-based formulation was initiated in [4]; ere, a local discontinuous Galerkin (LDG) metod was proposed for te low-frequency problem, covering te cases of eterogeneous media and topologically non-trivial domains. Te numerical experiments in [9] ave confirmed te p-convergence rates proved in [4] for smoot solutions, and indicate tat DG metods can be effective in a wide range of low-frequency applications were te bilinear forms are coercive. On te oter and, one of te main difficulties in te numerical solution of Maxwell s equations consists in dealing wit divergence-free constraints tat need to be imposed on te fields, especially in cases were te analytical solutions exibit strong singularities. Several approaces ave been proposed in te literature: we Department of Matematics and Computer Science, University of Leicester, Leicester LE 7RH, U, Paul.Houston@ mcs.le.ac.uk. Te researc of tis autor was supported by te EPSRC under grants GR/N430 and GR/R7665. Dipartimento di Matematica, Università di Pavia, Via Ferrata, 700 Pavia, Italy, perugia@ dimat.unipv.it. Matematics Department, University of Britis Columbia, 984 Matematics Road, Vancouver, BC V6T Z, Canada, scoetzau@ mat.ubc.ca.

2 P. HOUSTON, I. PERUGIA, D. SCHÖTZAU mention ere te (weigted) regularization metods studied in [, 4], te singular field approac of [6], and te Lagrange multiplier tecniques used in [9, 5, 6], for example. Te metods studied in [4, 9] are DG versions of te regularization approac of [] and, for singular solutions, were sown to suffer from similar drawbacks as teir conforming counterparts. A mixed discontinuous Galerkin approac was recently adopted in [5], were a stabilized interior penalty discretization was proposed for te ig-frequency time-armonic Maxwell equations. For smoot material coefficients, optimal convergence of te metod was proved by employing a duality approac, provided tat appropriate stabilization terms were included in te metod. In tis paper, we introduce and analyze a new mixed DG formulation for te Maxwell operator (consisting of te curl-curl operator subject to a divergence-free constraint). Altoug tis formulation is based on te same mixed approac as te one proposed in [5], ere te amount of numerical stabilization is drastically reduced. In particular, we abandon all te volume stabilization terms from [5] and acieve wellposedness of te formulation troug a suitable definition of te numerical fluxes. We present a numerical analysis of tis metod for piecewise constant material coefficients, and obtain a priori error bounds in te associated energy norm tat are optimal in te mes size, if bot te field and te Lagrange multiplier related to te divergence constraint are approximated wit piecewise polynomials of te same degree. Here, we consider bot te case were te underlying analytical solution is smoot and were only minimal regularity assumptions are assumed. Te metod proposed in tis paper is tested on a set of numerical examples tat confirm te convergence rates predicted in te teoretical analysis, for bot smoot and singular solutions on regular and irregular meses. Te metod is also tested witin an adaptive procedure on affine quadrilateral meses were anging nodes are introduced during te course of te refinement. Te numerical results indicate tat singularities present in te analytical solution are correctly captured by te proposed sceme. Te stability analysis of te mixed DG formulation is carried out along te following lines. Firstly, we rewrite te mixed system in an augmented form by introducing auxiliary variables, giving rise to a standard mixed saddle point problem wit nonconsistent forms. Ten, we establis coercivity of te curl-curl operator on a suitable kernel. Finally, we prove te inf-sup stability condition for te form related to te divergence constraint. Te proof of tis result makes use of ideas developed in [8] for te analysis of stabilized mixed metods and relies on a decomposition of te discontinuous Galerkin finite element space for te Lagrange multiplier into te direct sum of its largest conforming (stable) subspace and a corresponding complement. Te control over functions in te complement is ten ensured by a crucial norm equivalence property tat we establis by using an approximation result from [, Section.]. Te outline of te paper is as follows: in Section we introduce our mixed DG metod for te Maxwell operator. Our main teoretical results are te a priori error bounds presented in Section 3. Teir proofs are carried out in te following sections, were we introduce an auxiliary mixed formulation (Section 4), establis te continuity and stability properties of te forms involved (Section 5), and finally derive te actual error estimates (Section 6). Te numerical performance of te metod is tested in Section 7. Concluding remarks are presented in Section 8.. Model problem and discretization. In tis section, we introduce a mixed DG discretization of te curl-curl operator subject to a divergence-free constraint... Notation. We start by introducing te notation and function spaces tat will be used trougout tis paper. Given a bounded domain D in R or R 3, we

3 MIXED DG APPROXIMATION OF THE MAXWELL OPERATOR 3 denote by H s (D) te standard Sobolev space of functions wit integer or fractional regularity exponent s 0, and by s,d its norm. We also write s,d to denote te norms in te spaces H s (D) d, d =, 3. We set L (D) = H 0 (D). Furtermore, L (D) is te space of bounded functions on D. Given D R 3 and a positive weigt function w L (D), H(curl w ; D) and H(div w ; D) are te spaces of vector fields u L (D) 3 wit (wu) L (D) 3 and (wu) L (D), respectively, endowed wit teir corresponding grap norms. H(curl 0 w; D) and H(div 0 w; D) are te subspaces of H(curl w ; D) and H(div w ; D), respectively, of functions wit zero (weigted) curl and divergence, respectively. For w, we omit te subscript and write H(curl; D) and H(div; D), respectively. We denote by H 0 (D), H 0 (curl; D) and H 0 (div; D) te subspaces of H (D), H(curl; D) and H(div; D), respectively, of functions wit zero trace, tangential trace and normal trace, respectively... Model problem. Let be a bounded Lipscitz polyedron in R 3, wit n denoting te outward normal unit vector to its boundary Γ =. We assume tat te domain is simply-connected, and tat Γ is connected. We consider te following mixed model problem: find te vector field u and te scalar field p suc tat (µ u) ε p = j in, (εu) = 0 in, (.) n u = g on Γ, p = 0 on Γ. Here, te rigt-and side j L () 3 is an external source field, and te Diriclet datum g is a prescribed tangential trace wic we assume to belong to L (Γ) 3. Te coefficients µ = µ(x) and ε = ε(x) are real functions in L () tat satisfy 0 < µ µ(x) µ <, 0 < ε ε(x) ε <, a.e. x. (.) For simplicity, we assume tat µ and ε are piecewise constant wit respect to a partition of te domain into Lipscitz polyedra. Remark.. Problem (.) describes te principal operator of te time-armonic Maxwell equations in a eterogeneous insulating medium (i.e., wit electric conductivity σ = 0). Te coefficients µ and ε are te magnetic permeability and te electric permittivity of te medium, respectively. Te divergence constraint is incorporated by means of te Lagrange multiplier p; see, e.g., [5, 5, 6] and te references cited terein. Problem (.) is also a formulation of te magnetostatic problem in terms of te vector potential u and wit Coulomb s gauge u = 0 (ε in tis case). Remark.. In te discontinuous Galerkin context, te Diriclet boundary condition n u = g on Γ is enforced weakly by so-called interior penalty stabilization terms. In order to make tese terms well-defined for eac boundary face of a grid on, we use te regularity assumption g L (Γ) 3 wic is sligtly stronger tan te natural assumption for g. For less regular boundary data, te interior penalty terms need to be defined as suitable duality pairings over te wole boundary Γ. Setting V = {v H(curl; ) : (n v) Γ L (Γ) 3 } and Q = H0 (), te variational form of (.) is as follows: find u V, wit n u = g on Γ, and p Q suc tat a(u, v) + b(v, p) = j v dx, (.3) b(u, q) = 0 (.4)

4 4 P. HOUSTON, I. PERUGIA, D. SCHÖTZAU for all (v, q) H 0 (curl; ) Q, were te forms a and b are given, respectively, by a(u, v) = µ u v dx, b(v, p) = εv p dx. Well-posedness of te formulation (.3) (.4) follows from te standard teory of mixed problems [7], since a is bilinear, continuous and coercive on te kernel of b, and b is linear and continuous, and satisfies te inf-sup condition; see, e.g., [6] for details..3. Meses, finite element spaces and traces. Trougout, we consider sape regular and affine meses T tat partition te domain into tetraedra and/or parallelepipeds, wit possible anging nodes; we always assume tat te meses are aligned wit any discontinuities in te coefficients µ and ε. We denote by te diameter of te element T and set = max. An interior face of T is defined as te (non-empty) two-dimensional interior of +, were + and are two adjacent elements of T, not necessarily matcing. A boundary face of T is defined as te (non-empty) two-dimensional interior of Γ, were is a boundary element of T. We denote by F I te union of all interior faces of T, by F D te union of all boundary faces of T, and set F = F I F D. Given a nonnegative integer l and an element T, we define S l () as te space P l () of polynomials of degree at most l in, if is a tetraedron, or te space Q l () of polynomials of degree at most l in eac variable in, if is a parallelepiped. Similarly, for a face f F, we write S l (f) for te space P l (f) of polynomials of degree at most l in f, if f is a triangle, and te space Q l (f) of polynomials of degree at most l in eac variable in f, if f is a parallelogram. Ten, te generic finite element space of discontinuous piecewise polynomials is given by S l (T ) = {u L () : u S l () T }. For piecewise smoot vector- and scalar-valued functions v and q, respectively, we introduce te following trace operators. Let f F I be an interior face sared by two neigboring elements + and ; we write n ± to denote te outward normal unit vectors to te boundaries ±, respectively. Denoting by v ± and q ± te traces of v and q on ± taken from witin ±, respectively, we define te jumps across f by [v ] T = n + v + + n v, [v ] N = v + n + + v n and [q ] N = q + n + + q n, and te averages by {v } = (v + + v )/ and {q } = (q + + q )/. On a boundary face f F D, we set [v ] T = n v, [q ] N = q n, {v } = v and {q } = q..4. DG discretization. We wis to approximate problem (.) by discrete functions u and p in te finite element spaces V = S l (T ) 3 and Q = S l (T ), respectively, for a given partition T of, and an approximation order l. To tis end, we consider te DG metod: find (u, p ) V Q suc tat a (u, v) + b (v, p ) = f (v), (.5) b (u, q) c (p, q) = 0 (.6) for all (v, q) V Q, were te discrete forms a, b and c and te linear

5 MIXED DG APPROXIMATION OF THE MAXWELL OPERATOR 5 functional f are given, respectively, by a (u, v) = µ u v dx [u] T {µ v } ds F [v ] T {µ u } ds + a [u] T [v ] T ds + b [εu] N [εv ] N ds, F F F I b (v, p) = εv p dx + {εv } [p] N ds, F c (p, q) = c[p] N [q ] N ds, F f (v) = j v dx g µ v ds + a g (n v) ds, F D respectively. Here, denotes te elementwise operator. Te form a corresponds to te interior penalty discretization of te curl-curl operator [9, 5], wit te addition of a normal jump term; te form b discretizes te divergence operator in a DG fasion; and te form c is a stabilization form tat penalizes te jumps of p. Te parameters a, b and c are positive stabilization parameters tat will be cosen later on, depending on te mes size and te coefficients µ and ε. Note tat a similar discretization as been investigated in [5] for a time-armonic ig-frequency model of Maxwell s equations; we point out tat te additional stabilization forms tat ave been added tere become obsolete wit te analysis presented in tis paper. As in [4, Remark 3.] or [5, Proposition 4], it can be readily seen tat te analytical solution (u, p) V Q satisfies (.5) (.6) for all (v, q) V Q. Remark.3. All te interface contributions arising in te forms in (.5) (.6) can easily be obtained by rewriting te problem (.) as a first-order system and introducing so-called numerical fluxes in te sense of [5]. Tus, all te stabilization terms in (.5) (.6) are local, consistent and conservative. To see tis, we rewrite (.) as s µ u = 0, s ε p = j, (εu) = 0 in, subject to te boundary conditions n u = g and p = 0 on Γ. Ten we consider te following discretization: find (s, u, p ) V V Q suc tat s t dx s v dx µ t u dx + F D µ t û n ds = 0, v ŝ n ds + p (εv) dx (.7) p εv n ds = j v dx, εu q dx q εu n ds = 0 for all (t, v, q) V V Q, and for all elements in te partition T. In (.7), te traces of u, s, p and εu on are approximated by te numerical fluxes û = {u }, ŝ = {µ u } a[u ] T, p = {p } b[εu ] N, εu = {εu } c[p ] N,

6 6 P. HOUSTON, I. PERUGIA, D. SCHÖTZAU respectively, (tese definitions are for interior faces; tey must be suitably adapted for boundary faces). By integration by parts, te first equation of (.7) reads: s t dx = µ u t dx + µ t (u û ) n ds. (.8) As te numerical flux û is independent of s, te auxiliary variable s can be locally expressed in terms of u by inverting te local mass matrix s t dx in (.8). By substituting te resulting expression for s into te second equation of (.7), one obtains an elemental formulation for te unknowns u and p only. Finally, summing over all elements T gives te formulation (.5) (.6). We refer to [5] and [4] for furter details on te formalization of tis elimination process. 3. Main results. In tis section, we state our main results for te mixed DG metod in (.5) (.6); te proofs of tese a priori error bounds will be given in Sections 4, 5 and Stabilization parameters and DG-norms. We start by defining te stabilization parameters a, b and c appearing in (.5) (.6) and introduce te norms employed in te proceeding error analysis. To tis end, we first define te function in L (F ), representing te local mes size, as (x) = min{, }, if x is in te interior of for two neigboring elements in te mes T, and (x) =, if x is in te interior of Γ. Similarly, we define te functions m and e in L (F ) by m(x) = min{µ, µ } and e(x) = max{ε, ε }, if x is in te interior of, and m(x) = µ and e(x) = ε, if x is in te interior of Γ, wit µ and ε denoting te restrictions of µ and ε to te element, respectively. Wit tis notation, we coose te stabilization parameters as follows: a = α m, b = β e, c = γ e, (3.) were α, β and γ are positive parameters, independent of te mes size and te coefficients µ and ε. Furter, we set V () = (V H(div ε ; )) + V and Q() = Q + Q, and define: v V () = µ v 0, + m [v ]T 0,F + e [εv ]N 0,F, I v V () = ε v 0, + v V (), q Q() = ε q 0, + e [q ]N 0,F. We also introduce te space H s (T ) = {v L () : v H s (), T }, endowed wit te norm v s,t = T v s,. On te boundary, we define H s (F D) = { v L (F D) : v f H s (f), f F D }, equipped wit te norm v = s,f D f F v D s,f. 3.. Existence and uniqueness. Next, we sow tat te discrete problem in (.5) (.6) is uniquely solvable, provided tat α is sufficiently large. To tis end, we first recall te following well-known coercivity result, valid in view of te coice of a and te assumptions on te meses; see [5, 9, 5] for details. Lemma 3.. Tere exists a parameter α min > 0, independent of te mes size and te coefficients µ and ε, suc tat for α α min and β > 0 we ave a (v, v) C v V () for all v V, wit a constant C > 0 independent of te mes size and te coefficients µ and ε.

7 MIXED DG APPROXIMATION OF THE MAXWELL OPERATOR 7 Te condition α α min > 0 is a restriction tat is typically encountered wit symmetric interior penalty metods and may be omitted by using oter DG discretizations of te curl-curl operator, suc as te non-symmetric interior penalty or te LDG metod; see, e.g., [5, 4] for details. Proposition 3.. For α α min, β > 0 and γ > 0, te mixed DG metod (.5) (.6) possesses a unique solution. Proof. It is enoug to sow tat j = 0 and g = 0 imply u = 0 and p = 0. To tis end, take v = u in (.5) and q = p in (.6), ten subtract (.6) from (.5). Wit te coercivity of a in Lemma 3., it follows tat u = 0, [u ] T = 0 on F, [εu ] N = 0 on F I, and [p ] N = 0 on F, i.e., u H(curl 0 ; ) H 0 (curl; ), u H(div ε ; ) and p H0 (). Integrating by parts, (.6) becomes q (εu ) dx = 0, for any q Q, and ten, since ε is piecewise constant, (εu ) = 0. Terefore, u also belongs to H(div 0 ε; ), wic, owing to our assumptions on, implies tat u = 0; cf. [6, Section 4]. Equation (.5) becomes εv p dx = 0, for any v V, and ten p = 0. Since p = 0 on Γ, we conclude tat p = 0. For te rest of tis article, we sall assume tat te ypoteses on te stabilization parameters in te statement of Proposition 3. old A priori error estimates. First, we establis optimal error estimates for smoot solutions on possibly nonconforming meses subject to te following restrictions: (i) any interior face f F I as to be an entire elemental face of at least one of te two adjacent elements saring f; (ii) te number of interior faces contained in an elemental face is uniformly bounded wit respect to te mes size. Tis implies bounded variation of te local mes size, i.e., wenever and sare a common face and, we ave C C, for a constant C > 0, independent of te mes size. Te reason for tis restriction is related to te validity of te norm equivalence result of Teorem 5.3 below. Teorem 3.3. Let (u, p) be te analytical solution of (.) satisfying u H s+ (T ) 3 and p H s+ (T ) for a regularity exponent s >. Let (u, p ) be te mixed DG approximation obtained by (.5) (.6) on possibly nonconforming meses tat satisfy te restrictions (i) and (ii) above. Ten we ave te a priori error bound u u V () + p p Q() C min{s,l} [ u s+,t + p s+,t ], wit C > 0 depending on te bounds (.) on te coefficients µ and ε, te saperegularity and bounded variation properties of te mes, te stabilization parameters α, β and γ, and te polynomial degree l, but independent of te mes size. Wile te bound in Teorem 3.3 guarantees optimal convergence in te mes size wit respect to te polynomial degree used in te approximation, te smootness assumptions on te analytical solution are not minimal. In fact, for ε = µ = and a omogeneous Diriclet datum g 0, from te regularity results in [3], it follows tat one only as u H s () 3 and u H s () 3, for a regularity exponent s = s() > (te same actually olds for smoot coefficients ε and µ; see [5, Section.]). As far as te assumption p H s+ () is concerned, altoug it does not seem to old for general source terms j in L () 3, it is trivially satisfied in te pysically most relevant case of divergence-free source terms j, were p 0. For tese reasons, we state a second result under weaker smootness assumptions for te component u of te analytical solution. In order to do tis, we restrict ourselves to te case of conforming meses (i.e., meses wit no anging nodes); tis restriction is necessary, since te proof requires te use of H(curl; ) conforming projections.

8 8 P. HOUSTON, I. PERUGIA, D. SCHÖTZAU Note tat, since te boundary datum g is te tangential trace on Γ of a function in H(curl; ), its restriction to eac f F D is a two-dimensional vector field wic lies on te same plane as f. Hence, we can understand g as a function in L (Γ). Teorem 3.4. Let (u, p) be te analytical solution of (.) satisfying εu H s (T ) 3, µ u H s (T ) 3, p H s+ (T ) and g H s+ (F D), for a regularity exponent s >. Let (u, p ) be te mixed DG approximation obtained by (.5) (.6) on conforming meses. Ten we ave te a priori error bound u u V () + p p Q() C min{s,l} [ εu s,t + µ u s,t + p s+,t + g s+,f D wit C > 0 depending on te bounds (.) on te coefficients µ and ε, te saperegularity of te mes, te stabilization parameters α, β and γ, and te polynomial degree l, but independent of te mes size. Remark 3.5. Te numerical results reported in Section 7 sow tat, on a twodimensional L-saped domain, te above convergence rates are obtained also on nonconforming affine meses for te strongest corner singularities. Te mixed metod in (.5) (.6) enforces te divergence constraint in a weak sense only; neverteless, te convergence rate of te error in te (elementwise) divergence migt be of interest. Our last result addresses tis issue and proves a rate tat is of one order lower tan te error measured in te DG norm. Tis result is numerically observed to be sarp on conforming finite element meses, cf. Section 7. Teorem 3.6. Let (u, p) be te analytical solution of (.) and (u, p ) te mixed DG approximation obtained by (.5) (.6) on a possibly nonconforming mes satisfying te restrictions (i) and (ii) above. Ten we ave T (ε(u u )) 0, C [ [εu εu ] N F + e I [q q ] N 0, F ], wit C > 0 depending on te sape-regularity and bounded variation properties of te mes, te stabilization parameter γ, and te polynomial degree l, but independent of te mes size. Remark 3.7. Under te assumptions of bot Teorem 3.3 and Teorem 3.4, te estimate in Teorem 3.6 implies tat [ T (ε(u u )) 0, C min{s,l}, wit C > 0 independent of te mes size. Te proofs of Teorems 3.3, 3.4 and 3.6 are carried out in te next sections and concluded in Sections 6. and Auxiliary mixed formulation. In order to facilitate te error analysis, we rewrite te discrete formulation (.5) (.6) in a different (and perturbed) form, by introducing te jumps of p as auxiliary unknowns and by employing lifting operators as in [5, 4]. In tis way, te resulting bilinear forms ave suitable continuity and coercivity properties, so tat te metod can be analyzed using te classical teory of mixed finite metods. We begin by introducing te lifting operators L and M. For v belonging to V () and q Q(), we define L(v) V and M(q) Q by L(v) w dx = [v ] T {w } ds, F ] ], M(q) w dx = {w } [q ] N ds F

9 MIXED DG APPROXIMATION OF THE MAXWELL OPERATOR 9 for all w V. We ten define te perturbed forms ã (u, v) = µ u v dx L(u) (µ v) dx L(v) (µ u) dx + a [u] T [v ] T ds + b [εu] N [εv ] N ds, F F I b (v, p) = εv [ p M(p) ] dx. Note tat a = ã in V V and b = b in V Q, altoug tis is no longer true in V () V () and V () Q(), respectively. Next, we define te discrete space M = {λ L (F ) 3 : λ f S l (f) 3 f F }, endowed wit te norm η M = e η 0,F, and consider te following auxiliary mixed formulation: find (u, λ, p ) V M Q suc tat A (u, λ ; v, η)+b (v, η; p ) = f (v) (v, η) V M, (4.) B (u, λ ; q) = 0 q Q, (4.) wit forms A and B given, respectively, by A (u, λ; v, η) = ã (u, v)+ cλ η ds, B (v, η; p) = b (v, p) F F c[p] N η ds. F F Proposition 4.. Problem (4.) (4.) admits a unique solution (u, λ, p ) V M Q, wit (u, p ) te solution to (.5) (.6), and λ = [p ] N. Proof. By coosing test functions (0, η) in (4.), we ave cλ η ds = c[p ] N η ds η M. Since c is constant on eac f F, we ave tat λ = [p ] N. Ten, equations (4.) and (4.) coincide wit equations (.5) and (.6), respectively. Terefore, if (u, λ, p ) V M Q is a solution to (4.) (4.), ten λ = [p ] N and (u, p ) is (te unique) solution to (.5) (.6), wic proves uniqueness of te solution. Existence follows from uniqueness. Finally, we introduce te space W () = V () M and set (v, η) W () = v V () + η M, (v, η) W () = v V () + η M. Te proofs of Teorems 3.3, 3.4 and 3.6 are now carried out by analyzing te auxiliary mixed formulation (4.) (4.). In Section 5 we prove te continuity of A and B, te ellipticity of A on te kernel of B, as well as te inf-sup condition for B. In te proof of te inf-sup condition, we employ a norm equivalence property. Ten, te error estimates of Teorems 3.3, 3.4 and 3.6 are obtained in Section Continuity and stability. In tis section, we prove continuity properties of te forms A and B, te ellipticity of A on te kernel of B, as well as te inf-sup condition for B.

10 0 P. HOUSTON, I. PERUGIA, D. SCHÖTZAU 5.. Continuity properties. Te following continuity properties old. Proposition 5.. Tere exist constants a > 0 and a > 0, independent of te mes size and te coefficients µ and ε, suc tat A (u, λ; v, η) a (u, λ) W () (v, η) W (), (u, λ), (v, η) W (), B (v, η; q) a (v, η) W () q Q(), (v, η) W (), q Q(). Te linear functional f : V R on te rigt-and side of (4.) satisfies f (v) C [ ε j 0, + m g 0,Γ ] v V (), v V, wit a constant C > 0, independent of te mes size and te coefficients µ and ε. Proof. Proceeding as in [4, Proposition 4.] or [5, Proposition ], we ave te following stability estimates for L and M µ L(v) 0, C m [v ]T 0,F, ε M(q) 0, C e [q ]N 0,F, for any v V (), q Q(), wit a constant C > 0 tat is independent of te mes size and te coefficients µ and ε. Wit tese stability estimates, te continuity of A and B follows from te Caucy-Scwarz inequality and te coice of te stabilization parameters in (3.). Te continuity of f is obtained by using similar arguments; see [4, Corollary 4.5] for details. 5.. Ellipticity on te kernel. Define te discrete kernel er(b ) = {(u, λ) W : B (u, λ; p) = 0 p Q }. Proposition 5.. For α α min, β > 0, γ > 0, tere is a constant b > 0 independent of te mes size suc tat A (u, λ; u, λ) b (u, λ) W () for all (u, λ) er(b ). Proof. Trougout te proof, we denote by C any constant independent of te mes size and te coefficients µ and ε, and by C m any constant tat depends on te bounds (.) on te coefficients µ and ε, but is independent of te mes size. From Lemma 3., we immediately ave A (u, λ; u, λ) C (u, λ) W (), (u, λ) W. (5.) Now, fix (u, λ) er(b ) and let (z, ψ) be te solution of te auxiliary problem (µ z) ε ψ = εu (εz) = 0 in, subject to te boundary conditions n z = 0 and ψ = 0 on Γ. Tereby, µ z 0, + ε z 0, + (µ z) 0, + ε ψ 0, + ε ψ 0, C m ε u 0,. (5.) Set w = µ z; clearly, w H(curl; ). Terefore, from [6, Corollary 7.], tere exists w 0 H () 3 suc tat w 0 = w, w 0, C w H(curl;) C m ε u 0,. (5.3)

11 MIXED DG APPROXIMATION OF THE MAXWELL OPERATOR Multiplying te first equation of te auxiliary problem by u and integrating by parts over eac element, we get ε u 0, = w 0 u dx w 0 [u] T ds+ ψ (εu) dx {ψ }[εu] N ds. F F I Since (u, λ) er(b ), we ave B (u, λ; ψ ) = 0 for any ψ Q and obtain, from integration by parts and te fact tat [ψ ] N = 0 on F, ε u 0, = w 0 u dx w 0 [u] T ds + (ψ ψ ) (εu) dx F {ψ ψ }[εu] N ds e λ [ψ ψ ] N ds. F I F Using (.) and (5.3), we ave w 0 u dx C m w 0, µ u 0, C m ε u 0, u V (). Furtermore, using trace inequalities, (.) and (5.3), we obtain ( w 0 [u] T ds C ) µ w 0 0, m [u]t 0,F F T C m w 0, m [u]t 0,F C m ε u 0, u V (). For te oter terms, we coose ψ as te L -projection of ψ on Q. Since ε is piecewise constant, we ave (ψ ψ ) (εu) dx = 0. Ten, using te Caucy- Scwarz inequality, te definition of e and standard approximation properties of te L -projection, we obtain F I Similarly, ( {ψ ψ }[εu] N ds C C ( T ε ψ 0, + ε ψ 0, T ε ψ ψ 0, ) u V () ) u V () C m ε u 0, u V (). F ( e λ [ψ ψ ] N ds C T ε ψ ψ 0, C m ε u 0, λ M. ) ( F ) e λ ds Te above computations sow tat ε u 0, C m (u, λ) W (). Combining tis wit (5.) and te definition of (u, λ) W () completes te proof Inf-sup condition. In tis section, we prove te inf-sup condition for te form B. Our proof is inspired by recent ideas from [8] used in te analysis of stabilized mixed metods. We will make use of te following crucial norm equivalence result. To tis end, let Q c be te subspace Q H0 () of Q, and let Q be te ortogonal

12 P. HOUSTON, I. PERUGIA, D. SCHÖTZAU complement in Q of Q c, wit respect to te norm Q(). We observe tat q Q = e [q ] N 0,F is a norm on Q. Indeed, if q Q and q Q = 0, ten q Q Qc = {0}. Te norms Q and Q() are equivalent in Q. Teorem 5.3. Tere are positive constants C and C, independent of mes size and µ and ε, suc tat C q Q() q Q C q Q() for any q Q. Proof. Step : Te following approximation result olds: for any q Q, inf ε (q q c ) 0, C e [q ]N 0,F, (5.4) q c Q c wit a constant C > 0 independent of te mes size and te coefficients µ and ε. Tis result as been proved in [, Teorem. and Teorem.3] for simplicial meses. Te proof tere can be easily generalized to te meses considered in tis paper and readily gives te independence of te constant on te coefficients µ and ε; we refer to [8, Appendix A] for tese tecnical details. Step : Te inequality on te rigt and side of te norm equivalence is trivially satisfied wit C =. To sow te bound on te left-and side, let P : Q Q denote te Q() ortogonal projection. For q Q, we ten ave P q Q() = inf q Q c q q Q() C P q Q. Here, we ave used properties of ortogonal projections, te approximation result (5.4), te fact tat [q ] N = [P q ] N, and te definition of Q. Since P is surjective, te equivalence follows. Our main result of tis section is te following inf-sup condition. Proposition 5.4. We ave inf 0 q Q B (v, ν; q) sup κ > 0, 0 (v, ν) W q Q() (v, ν) W () for a constant κ, independent of te mes size and te coefficients µ and ε. Proof. Fix 0 q Q arbitrary and consider its Q()-ortogonal decomposition as q = q 0 q, wit q 0 Q c and q Q. By coosing v 0 = q 0 V H(curl 0 ; ) H 0 (curl; ), we ave Furtermore, by te definition of e, B (v 0, 0; q 0 ) = ε q0 0, = q 0 Q(). (5.5) (v 0, 0) W () = e [εv0 ] N 0,F + ε I v0 0, C q0 0, C ε q0 0, = C q 0 Q(). (5.6) T ε q 0 0, + ε Here, we ave used te discrete trace inequality ϕ 0, C ϕ 0,, valid for polynomials ϕ S l (), wit a constant C > 0 tat is independent of te local mes size. Next, setting ν = [q ] N, gives B (0, ν ; q )=γ e [q ] N ds γc q Q(), (0, ν ) W () C q Q(), (5.7) F were C and C are te constants in te norm equivalence of Teorem 5.3. Ten, we set (v, ν) = (v 0, 0) + δ(0, ν ), wit a parameter δ > 0 still at our disposal. Since B (0, ν ; q 0 ) = 0, we obtain from (5.5) and (5.7), B (v, ν; q) = B (v 0, 0; q 0 ) + B (v 0, 0; q ) + δb (0, ν ; q ) q 0 Q() + δγc q Q() B (v 0, 0; q ).

13 MIXED DG APPROXIMATION OF THE MAXWELL OPERATOR 3 Combining te continuity of B (see Proposition 5.) wit a weigted Caucy-Scwarz inequality and (5.6), we obtain for any ζ > 0, B (v 0, 0; q ) Cζ v 0 V () + C ζ q Q() Cζ q 0 Q() + C ζ q Q(). Hence, by suitably coosing δ and ζ, we ave B (v, ν; q) κ [ q0 Q() + q Q()] = κ q Q(), (5.8) were we used te ortogonality of te decomposition of q. From (5.6) and (5.7), (v, ν) W () κ q Q(). (5.9) Te constants κ and κ in (5.8) and (5.9) are independent of te mes size and te coefficients µ and ε. Te proposition follows from (5.8) and (5.9), wit κ = κ /κ. 6. Error estimates. In tis section, we prove te error estimates stated in Teorems 3.3, 3.4 and Abstract error estimates. We start by deriving abstract error bounds. To tis end, for te analytical solution (u, p) to (.3) (.4), we define te residuals R (u, p; v, ν) = A (u, 0; v, ν) + B (v, ν; p) f (v) and R (u; q) = B (u, 0; q) for all (v, ν) W and q Q, and set R R (u, p; v, ν) (u, p) = sup, R R (u; q) 0 (v, ν) W (v, ν) (u) = sup. W () 0 q Q q Q() In te following teorem we present abstract error estimates for our DG metod. Tese error bounds are obtained by extending te standard conforming mixed finite element teory [7] to te setting considered ere, and taking into account te residual terms arising from te nonconsistency of te perturbed formulation. Teorem 6.. Tere exists positive constants C, suc tat [ (u u, λ ) W () C max{, b } [ p p Q() C inf v V u v V () ] + inf p q Q() + R (u, p) + q Q R (u), ]. inf q Q p q Q() + (u u, λ ) W () + R (u, p) Here, b, te ellipticity constant from Proposition 5., depends on te bounds in (.), but is independent of te mes size. Te constants C depend on te continuity constants a and a in Proposition 5., and te inf-sup constant κ from Proposition 5.4, but are independent of te mes size, µ and ε. Proof. By te triangle inequality and te definition of (, ) W (), we ave (u u, λ ) W () (u v, η) W () + (v u, η λ ) W (), (6.) for any (v, η) W. First, we take (v, η) er(b ). Since (v u, η λ ) er(b ), employing te ellipticity property of Proposition 5. and te definition of R, we ave b (v u, η λ ) W () A (v u, η λ ; v u, η λ ) = A (v u, η; v u, η λ ) B (v u, η λ ; p q) + R (u, p; v u, η λ ),

14 4 P. HOUSTON, I. PERUGIA, D. SCHÖTZAU for any q Q. From te continuity properties of Proposition 5., te definition of te norm (, ) W () and (6.), we ave ( (u u, λ ) W () + a b Next, we prove tat + a b ( inf (u v, η) W () (v, η) er(b ) ) inf (v, η) er(b ) (u v, η) W () inf p q Q() + (6.) q Q b R (u, p). + a κ ) inf (v, η) W (u v, η) W () + κ R (u). (6.3) To tis end, let (v, η) be any element of W, and consider te following problem: find (w, ν) W suc tat B (w, ν; q) = B (u v, η; q) R (u, q) q Q. (6.4) Problem (6.4) admits solutions in W tat are unique up to elements in te kernel of B. Te discrete inf-sup condition of Proposition 5.4 guarantees te existence of a solution (w, ν) satisfying (w, ν) W () [ B (u v, η; q) R ] (u, q) sup + sup κ q Q q Q() q Q q Q() a κ (u v, η) W () + κ R (u), (6.5) were we ave used te continuity of B, te definition of te norm (, ) W (), and te definition of R. From (6.4), B (w + v, ν + η; q) = 0, for any q Q, so tat (w + v, ν + η) er(b ). Terefore, since (u (v + w), η + ν) W () (u v, η) W () + (w, ν) W (), for any (v, η) W, taking into account (6.5), we obtain (6.3). Tis, togeter wit (6.), yields [ (u u, λ ) W () C max{, b } inf (u v, η) W () (v, η) W ] + inf p q Q() + R (u, p) + q Q R (u), were te constant C depends on a, a, and κ. Coosing η = 0, gives te error bound for (u u, λ ). We now turn to te bound for p p. Again by te triangle inequality, we ave for any q Q. Since p p Q() p q Q() + q p Q(), (6.6) A (u u, λ ; v, η) + B (v, η; p q) + B (v, η; q p ) = R (u, p; v, η),

15 MIXED DG APPROXIMATION OF THE MAXWELL OPERATOR 5 for any (v, η) W, te discrete inf-sup condition of Proposition 5.4 gives q p Q() κ B (v, η; q p ) sup (0, 0) (v, η) W (v, η) W () = A (u u, λ ; v, η) B (v, η; p q) + R (u, p; v, η) sup κ (0, 0) (v, η) W (v, η) W () a κ (u u, λ ) W () + a κ p q Q() + κ R (u, p). Tis, togeter wit (6.6), gives te bound for p p. 6.. Proof of Teorem 3.3 and Teorem 3.4. We are now ready to prove our main results, by making explicit te abstract error estimates of Teorem 6.. First, we derive bounds on te residuals; we note tat te residuals are optimally convergent on possibly nonconforming meses under minimal smootness assumptions as in Teorem 3.4, tereby covering bot te cases of Teorems 3.3 and 3.4. Proposition 6.. Let T be a possibly nonconforming mes satisfying te restrictions (i) and (ii) in Section 3.3. Assume te analytical solution (u, p) of (.) satisfies εu H s (T ) 3 and µ u H s (T ) 3, for s >. Ten we ave R (u, p) + R (u) C min{s,l+}[ εu s,t + µ u s,t ], wit a constant C > 0, independent of te mes size. Proof. First, for v V, η M, we ave R(u, p; v, η) = {µ u Π V (µ u) } [v ] T ds, F wit Π V denoting te L -projection onto V. Tis can be easily proved by employing integration by parts, te properties of te L -projection and taking into account te first equation in (.). From te Caucy-Scwarz inequality and standard approximation properties, we obtain R (u, λ, p; v, η) C ( T µ u Π V (µ u) 0, C min{s,l+} µ u s,t (v, η) W (). Furtermore, for q Q, R F (u; q) = {εu Π V (εu) } [q ] N ds. Hence, R (u; q) C ( T εu Π V (εu) 0, ) ) (v, η) W () q Q() C min{s,l+} εu s,t q Q(). Tis completes te proof (te constant C in tese estimates actually depends on te bound (.) on te coefficients µ and ε). Next, we prove te result of Teorems 3.3 and 3.4.

16 6 P. HOUSTON, I. PERUGIA, D. SCHÖTZAU Proof of Teorem 3.3: Let (u, p, λ ) be te solution of te auxiliary mixed system in (4.) (4.). We apply Teorem 6. and bound all te terms in te abstract error bounds tere. To tis end, we set v = Π V u and q = Π Q p wit Π V and Π Q denoting te L projections onto V and Q, respectively. Standard approximation properties, togeter wit te bounded variation of te mes size, give u Π V u V () C min{s,l} u s+,t. Te additional smootness assumption on u made in tis case is required for te estimate of te term containing te tangential jumps. Similarly, we ave p Π Q p Q() C min{s,l} p s+,t. Te constants C in te previous estimates depend on te bounds (.) on te coefficients µ and ε. Inserting tese estimates, togeter wit te residual estimates of Proposition 6., in Teorem 6. gives te result. Proof of Teorem 3.4: Let (u, p, λ ) be te solution of te auxiliary mixed system in (4.) (4.). As in te proof of Teorem 3.3, we apply Teorem 6.. On conforming meses, we can coose v = Π curl u H(curl; ) as te standard conforming Nédélec interpolant of u of te second type; see [3]. Tereby, we ave [v ] T = 0 on F I and from te approximation results proved in [] for tetraedra, but also valid for parallelepipeds, we get u Π curl u 0, + (u Π curl u) 0, C min{s,l} [ ] u s, + u s,. Hence, ε (u Πcurl u) 0, + µ (u Πcurl u) 0, C min{s,l} [ εu s,t + µ u s,t ], (6.7) wit C also depending on te bounds (.) on te coefficients µ and ε. On a boundary face f F D, te tangential field n Π curlu coincides wit Π div (n u), were Π div is te two-dimensional H(div)-conforming Nédélec interpolation operator of te second type. In particular, Π div reproduces polynomial tangential fields of degree l on f; see [3] for details. From a standard scaling arguments, we obtain n (u Π curl u) 0,f C min{s,l}+ f g s+,f. Terefore, summing over all faces yields m (n u Πcurl (n u)) 0,F D C min{s,l} g s+,f D. (6.8) Moreover, since u H(div 0 ε ; ), we ave for an interior face f, sared by f and f, e [ε(u Πcurl u)] N 0,f C min{s,l} [ u s,f f + u s, f f ]. Tis bound follows from [5, Lemma 3]. Hence, e [ε(u Πcurl u)] N 0,F I C [ εu s,t + µ u s,t ]. (6.9) Moreover, coosing q = Π H p in Teorem 6., were Π H is a standard H -projector, we ave [q ] N = 0 on F and p Π H p Q() C min{s,l} p s+,t. (6.0)

17 MIXED DG APPROXIMATION OF THE MAXWELL OPERATOR 7 Combining (6.7) (6.0) wit te residual estimates in Proposition 6. yields inf u v V () + inf p q Q() + R (u, p) + R (u) v V q Q C min{s,l} [ εu s,t + µ u s,t + p s+,t + g s+,f D wit C also depending on te bounds (.) on te coefficients µ and ε. Combining tese estimates wit te residual estimates of Proposition 6. gives te result Proof of Teorem 3.6. To prove Teorem 3.6, we first recall tat B (u, λ ; q) = 0 q Q. Note tat λ = [p] N, cf. Proposition 4.; tereby, for an element T, we ave εu q dx + {εu } (qn ) ds γ e [p ] (qn ) ds = 0 (6.) for all q S l (). Employing integration by parts and te identity (6.), we get, after some elementary manipulations, te following (εu )q dx = εu q dx + εu (qn ) ds = [εu ] N q ds + γ e [p ] (qn ) ds, 0 for q S l (), were 0 = F I. Te Caucy-Scwarz inequality gives ) (εu )q dx C q 0, [εu ] ( e N ds + [p ] N ds, 0 wit a constant C > 0, independent of te mes size. Here, we used sape-regularity and bounded variation properties of te mes. Employing te discrete trace inequality q 0, C q 0, for all q S l (), and te caracterization we obtain (εu ) 0, = sup (εu )q dx, q S l () q 0, ( (εu ) 0, C [εu ] N ds + 0 ) e [p ] N ds. Summing over all elements and taking into account tat te analytical solution satisfies [εu] N = 0 on F I and [p] N = 0 on F completes te proof. 7. Numerical experiments. In tis section, we present a series of numerical experiments to illustrate te a priori error estimates derived for te mixed DG metod introduced in Section. Here, we restrict ourselves to two-dimensional model problems wit constant coefficients µ ε. In tis case, by identifying two-dimensional vector fields u(x, y) = (u (x, y), u (x, y)) in R wit teir tree-dimensional extensions u(x, y, z) = (u (x, y, 0), u (x, y, 0), 0) in R 3, we deduce tat ( u) = ( y ( u x u y ), x ( u x u y )). ],

18 8 P. HOUSTON, I. PERUGIA, D. SCHÖTZAU (a) (b) Fig. 7.. Example. (a) Quadrilateral mes (i); (b) Quadrilateral mes (ii). u u V () PSfrag replacements Squares Quads (i) Quads (ii) l = l = l = 3 l = 4 Fig. 7.. Example. Convergence of u u V () wit refinement. On te boundary, we ave n u = u t, were t is te counterclockwise oriented tangential unit vector, i.e., if n = (n, n ), ten (t, t ) = ( n, n ). Hence, te Diriclet boundary datum given in (.) is a scalar function g. Similarly, te tangential jumps are scalar quantities defined as [u] T = u + t + + u t. We sall restrict our attention to meses consisting of quadrilateral elements only. In tis case te finite element space S l (T ) is constructed by mapping te reference element ˆ = (, ) onto eac element in te computational mes T, via te standard bilinear mapping F : ˆ. Tereby, discrete functions, restricted to a given element, are defined as u F = û, were û Q l ( ˆ). We point out tat meses obtained wit non-affine mappings F are not rigorously covered by our analysis and te underlying stability and approximation properties need to be furter investigated; see [4] for related work on te approximation properties of bilinearly mapped quadrilateral elements. Finally, we note tat trougout tis section we select te constants appearing in te stabilization parameters defined in (3.) as follows: α = 0 l, β = and γ =. We remark tat te dependence of α on te polynomial degree l as been formally cosen in order to guarantee te coercivity property in Lemma 3. of te underlying DG form a independently of l, cf. [9], for example. 7.. Example. Here, we let be te L-saped domain (, ) \[0, ) (, 0]; furter, we coose j and g so tat te analytical solution to te two-dimensional

19 MIXED DG APPROXIMATION OF THE MAXWELL OPERATOR 9 p p Q() PSfrag replacements Squares Quads (i) Quads (ii) l = l = l = 3 l = 4 Fig Example. Convergence of p p Q() wit -refinement. analogue of (.) wit µ ε is given by u exp(x)(y cos(y) + sin(y)) u = exp(x)y sin(y) p sin(π(x )/) sin(π(y )/) ; (7.) tis is a variant of te model problem considered in [9]. We investigate te asymptotic beavior of te errors of te mixed DG metod (.5) (.6) on a sequence of successively finer square and quadrilateral meses for different values of te polynomial degree l. In eac case we consider two types of quadrilateral meses wic are constructed from a uniform square mes by (i) randomly perturbing eac of te interior nodes by up to 0% of te local mes size, cf. Figure 7.(a); (ii) randomly splitting eac of te interior nodes by a displacement of up to 0% of te local mes size, cf. Figure 7.(b). Te latter meses are constructed so tat all te nodes in te interior of are irregular (i.e., anging), cf. [0]. In Figure 7. we first present a comparison of te DG norm V () of te error in te approximation to u wit te mes function for l 4. For consistency, u u V () is plotted against u for eac mes type, were u denotes te mes size of te uniform square mes; tis ensures tat a fair comparison between te error per degree of freedom for eac mes type can be made. Here, we observe tat u u V () converges to zero, for eac fixed l, at te rate O( l ) as te mes is refined, tereby confirming Teorem 3.3. In particular, we observe tat wile te error on te square mes is smaller tan on te randomly generated quadrilateral mes (i), as we would expect, te error is consistently smaller wen te irregular quadrilateral mes is employed. As in [0], we attribute tis improvement in u u V () to te increase in inter-element communication on te meses (ii); wen no anging nodes are present in te mes, elements may only communicate wit teir four immediate neigbors. On te oter and, on irregular meses elements may now communicate wit all of teir neigbors wic sare a common node, cf. [0]. Secondly, in Figure 7.3 we plot te DG norm Q() of te error in approximating p by p as te mes size tends to zero. As for te approximation to u, we again observe tat p p Q() converges to zero, for eac fixed l, at te rate O( l ) as te mes is refined, cf. Teorem 3.3. However, in contrast to te approximation to u, bot te conforming and nonconforming quadrilateral meses lead to a sligt degradation in te

20 0 P. HOUSTON, I. PERUGIA, D. SCHÖTZAU 0 0 u u 0, PSfrag replacements l = l = l = 3 l = Squares Quads (i) Fig Example. Convergence of u u 0, wit -refinement. u u 0, PSfrag replacements l = l = l = 3 l = Quads (ii) Fig Example. Convergence of u u 0, wit -refinement. size of te error in te approximation to p for eac mes and eac polynomial degree employed; toug, in almost all cases te error in te numerical solution computed on te meses (ii) was observed to be sligtly smaller tan te corresponding quantity computed on te meses (i). Tereby, te increase in inter-element communication arising wen te meses (ii) are employed no longer leads to te improvement in te size of te approximation error observed above for u as well as in [0]. Te increase in te quality of te numerical approximation u to u wen te nonconforming meses (ii) are employed becomes even more apparent wen te error u u is measured in terms of te L () norm. To tis end, in Figure 7.4, we first plot u u 0, against for l 4 using te uniform square and randomly generated quadrilateral meses (i). As predicted by Teorem 3.3, we observe tat u u 0, converges to zero, for eac fixed l, at te rate O( l ) as te mes is refined. Wile tis rate of convergence is one order less tan we would expect wen using discontinuous piecewise polynomials of degree at most l in eac coordinate direction, te numerical results clearly verify te sarpness of te a priori error analysis. However, in contrast, wen te nonconforming quadrilateral meses (ii) are

21 MIXED DG APPROXIMATION OF THE MAXWELL OPERATOR e 0, PSfrag replacements l = l = l = 3 l = Squares Quads (i) Fig Example. Convergence of e 0, wit -refinement. e 0, PSfrag replacements l = l = l = 3 l = Quads (ii) Fig Example. Convergence of e 0, wit -refinement. employed, te order of convergence increases by a full power of ; tereby, in tis case u u 0, now converges to zero, for eac fixed l, at te rate O( l+ ) as tends to zero, cf. Figure 7.5. Analogous beavior is also observed wen te L () norm of te error in te approximation to te divergence of u is computed. Indeed, from Figure 7.6, we observe tat e 0, converges to zero at te rate O( l ) as tends to zero, wen te uniform and randomly generated quadrilateral meses (i) are employed, tereby confirming Teorem 3.6 and Remark 3.7. On te oter and, wen te nonconforming quadrilateral meses (ii) are employed, tis rate of convergence increases to O( l+ ) as tends to zero, cf. Figure 7.7. As a final remark, we note tat on all te meses employed, te L () norm of te error in te approximation to p converges to zero, for eac fixed l, at te (optimal) rate O( l+ ) as te mes is refined. As for te DG norm of p p, bot te conforming and nonconforming quadrilateral meses lead to a sligt degradation in te size of p p 0, for eac mes and eac polynomial degree employed, toug te error in te numerical solution computed on te meses (ii) was observed to be sligtly smaller tan te corresponding quantity computed on te meses (i); for brevity, tese results ave been omitted.

22 P. HOUSTON, I. PERUGIA, D. SCHÖTZAU l = l = l = 3 Elements u u V () k u u V () k u u V () k 5.987e e e e e e e e e e e e e e e Table 7. Example. Convergence of u u V () on uniform square meses wit refinement. l = l = l = 3 Elements p p Q() k p p Q() k p p Q() k 8.74e e e e e e e e e e e Table 7. Example. Convergence of p p Q() on uniform square meses wit refinement. 7.. Example. In tis second example, we investigate te performance of te mixed DG metod (.5) (.6) for a problem wit a corner singularity in u. To tis end, we again let be te same L-saped domain as in te first example; ere, we set j = 0 and g is cosen so tat te analytical solution u to te two-dimensional analogue of (.) wit µ ε is given, in terms of te polar coordinates (r, ϑ), by u(x, y) = S(r, ϑ), were S(r, ϑ) = r /3 sin(ϑ/3); tereby, p 0. Te analytical solution u contains a singularity at te corner located at te origin; of ; ere, we only ave u H /3 ε (), ε > 0. In tis example, let us first confine ourselves to uniform square meses; we sall return to te more general meses considered in te previous example later. To tis end, in Tables 7. and 7. we present a comparison of te DG norms of te error in te approximation to bot u and p, respectively, wit te mes function on a sequence of uniform square meses for l 3. In eac case we sow te number of elements in te computational mes, te corresponding DG norm of te error and te computed rate of convergence k. Here, we observe tat (asymptotically) bot u u V () and p p Q() converge to zero at te optimal rate O( min(/3 ε,l) ), as tends to zero, predicted by Teorem 3.4. Next, in Table 7.3 we present a comparison of te L () norm of te error in te numerical approximation to u wit. On te basis of Teorem 3.4, we expect tat u u 0, sould tend to zero at te rate O( min(/3 ε,l) ) as tends to zero. However, from Table 7.3, we observe tat for l =,, 3, te rate of convergence of te L () norm of te error in te approximation to u is sligtly iger tan predicted; altoug, asymptotically, we expect tese convergence rates to slowly tend to te optimal one. Additionally, in Table 7.4 we sow te convergence of e 0, wit respect to ; ere, we again observe tat, asymptotically, te rate of convergence tends to te one predicted in Teorem 3.6, cf. Remark 3.7. Finally, in Table 7.5 we sow p p 0, for l =,, 3 based on employing uniform square meses. In comparison