Portland State University PDXScolar Matematics and Statistics Faculty Publications and Presentations Fariborz Masee Department of Matematics and Statistics 200 A Class of Discontinuous Petrov Galerkin Metods. Part I: Te Transport Equation Leszek Demkowicz University of Texas at Austin Jay Gopalakrisnan Portland State University, gjay@pdx.edu Let us know ow access to tis document benefits you. Follow tis and additional works at: ttps://pdxscolar.library.pdx.edu/mt_fac Part of te Applied Matematics Commons, and te Matematics Commons Citation Details Demkowicz, Leszek and Gopalakrisnan, Jay, "A Class of Discontinuous Petrov Galerkin Metods. Part I: Te Transport Equation" (200). Matematics and Statistics Faculty Publications and Presentations. 55. ttps://pdxscolar.library.pdx.edu/mt_fac/55 Tis Post-Print is brougt to you for free and open access. It as been accepted for inclusion in Matematics and Statistics Faculty Publications and Presentations by an autorized administrator of PDXScolar. For more information, please contact pdxscolar@pdx.edu.
A CLASS OF DISCONTINUOUS PETROV-GALERKIN METHODS. PART I: THE TRANSPORT EQUATION L. DEMKOWICZ AND J. GOPALAKRISHNAN Abstract. Considering a simple model transport problem, we present a new finite element metod. Wile te new metod fits in te class of discontinuous Galerkin (DG) metods, it differs from standard DG and streamline diffusion metods, in tat it uses a space of discontinuous trial functions tailored for stability. Te new metod, unlike te older approaces, yields optimal estimates for te primal variable in bot te element size and polynomial degree p, and outperforms te standard upwind DG metod.. Introduction We introduce a new Petrov-Galerkin metod for advective problems. Wile it belongs in te class of discontinuous Galerkin (DG) metods, unlike te standard upwind DG metod, we are able to prove optimal and p error estimates in te L 2 -norm for our discrete solution on general meses (were, as usual, is te mes size and p is te polynomial degree). Te metod includes a separate outflux approximation on element interfaces and a space of non-standard test functions designed for stability. Te boundary value problem tat is te subject of tis paper is posed on a polyedral domain Ω. Given f and g, we need to find a finite element approximation to te solution u of β u = f on Ω, (a) u = g on in Ω. (b) We only consider te case of constant β in tis paper (but extensions are possible, as mentioned in Section 5). Te inflow boundary in Ω appearing in (), is defined, letting n denote te unit outward normal, by in Ω = { x Ω : β n( x) < 0}, (2) i.e., in Ω denotes te global inflow boundary. Wile non-finite-element numerical tecniques can be designed for tis problem (e.g. te metod of caracteristics), we aim for finite elements because of its versatility in andling complicated domains as well as certain regular and singular perturbations of te above problem. A regular perturbation of () is β u + α( x)u = f. (3) A singular perturbation of () is obtained by te addition of a small viscosity term wit second derivatives. Tis is arder to analyze. Witin te domain of finite element metods for (), tere are two broad categories (see [] for a review). One is te very popular Te work of te first autor was supported by DOE troug Predictive Engineering Science (PECOS) Center at ICES (PI: Bob Moser), and by a researc contract wit Boeing. Te second autor was supported in part by te National Science Foundation under grants 073833, 069080, and an Oden fellowsip at ICES.
2 L. DEMKOWICZ AND J. GOPALAKRISHNAN streamline diffusion metod [3] and its descendants. Te oter category is composed of DG metods. Since our contribution fits in te latter, we sall now review previous works in tis category in detail. Te well known first papers proposing and analyzing te original DG metod for () are [4, 5, 8]. To distinguis tis metod from our DG metod, we will call te original DG metod te upwind DG metod and denote it by UDG, wile we call ours te discontinuous Petrov-Galerkin metod and denote it by DPG. It is proved in [5] tat if u is te UDG approximation, ten (for a fixed p) it satisfies u u L 2 (Ω) O( s ) for some s p + dictated by te regularity of te exact solution. Tis result was improved by [4] werein it was sown tat te rate of convergence is in fact O( s /2 ). In bot cases convergence wit respect to p was not studied. Even if we set aside te p-convergence issue, notice tat bot te results are suboptimal in, as te best approximation error of te finite element space is O( s ). For some special classes of meses owever, many autors ave observed (and proved) te optimal rate of convergence of te UDG metod [8, 9] wit respect to. Noneteless, on general meses, te suboptimal rate of convergence cannot be improved, as sown by a numerical example in [7] using a particular quasiuniform mes and a smoot exact solution. To express te sentiment of many, we quote from [8] tat te mecanisms tat induce te loss of /2 in te order of convergence of te L 2 -norm of te error are not very well known yet. An p analysis of te UDG sceme was first provided in [3]. Tey considered te regular perturbation (3) under te assumption tat 0 < c 0 α( x) x Ω (4) Because of tis assumption, tey are able to control te L 2 (Ω)-norm of te solution. Tey also introduced a stabilization parameter into te original upwind DG metod. A few years later, te paper [2] extended te results of [3] in several directions, providing a unified teory for an p version of te streamline diffusion metod, as well as te upwind DG metod. Teir analysis did not assume (4), rater tey let te advection vector β depend on x and assumed 0 < c 0 2 β( x) + α( x) x Ω, (5) as a consequence of wic tey ave stability in c 0 u L 2 (Ω). (Note tat for te case we intend to study (), te rigt and side of (5) evaluates to zero, so (5) does not old.) Bot papers analyzed te stabilized version of te upwind DG metod and bot relied on te proper coice of te stabilization parameter. Wile te results of [2] are optimal in p, tose of [3] are suboptimal in p. In all tese works, te L 2 (Ω)-rate of convergence of te error wit respect to remained suboptimal by /2. In contrast, our results do not exibit tis suboptimality, nor do we add any stabilization parameter. We believe our metod is te first in te finite element (not just DG) family of metods for te transport equation wic as provably optimal convergence rates on very general meses. Te design of our metod is guided by a generalization of Céa lemma due to Babuška [, 5]. We only need a simple version of te result, wic we now describe using te following notations (all our spaces are over R): Let X, X X, and V be Banac spaces and let a (, ) be a bilinear form on X V. Suppose te exact solution U X satisfies and te
DPG METHOD 3 discrete solution U X satisfies a (U U, v ) = 0 for all v V. (6) If te bilinear form is continuous in te sense tat tere is a C > 0 suc tat a (w, v ) C w X v V for all w X, v V (7) and also te inf-sup condition, i.e., tere is a C 2 > 0 suc tat C 2 w X sup v V a (w, v ) v V for all w X, (8) ten, as is well known, te following teorem can be formulated: Teorem.. Under te above setting, we ave te following error estimate: U U X ( + C ) inf U w X. C 2 w X Proof. Te argument is simple and standard: U U X = U w X + w U X U w X + C 2 U w X + C 2 a (w U, v ) sup by (8), v V v V a (w U, v ) sup by (6), v V v V U w X + C C 2 w U X by (7). Tis finises te proof. Many refinements and improvements of suc a teorem are known. But our purpose in going troug te above simple argument is to clearly sow tat te test space need not ave approximation properties. Hence, in designing Petrov-Galerkin metods, wile we must coose trial spaces wit good approximation properties, we may design test spaces solely to obtain good stability properties. Tis will be our guiding principle in designing our metod. In fact, te test spaces we propose sortly can ave discontinuities inside te mes elements. Many researcers ave put te above principle to good use. In fact, even te abbreviation we use for our new metod DPG metod, as been previously used [4, 6] for oter metods. Te teme in tese works is te searc for stable test spaces using bubbles or oter polynomials. Our test space functions, in contrast, need not be polynomial on an element, and indeed, need not even be continuous. We are also not te first to consider suc functions wit discontinuities witin a finite element. Suc elements are routinely used in X-FEM and similar metods [2] for difficult simulations like crack propagation. However, we use discontinuities solely for stability purposes, and solely in test spaces. Our trial spaces, being standard polynomial spaces, possess provably good approximation properties. Our metod also introduces a new flux unknown on te element interfaces. Tis is in line wit te recent developments on ybridized DG (HDG) metods [9]. HDG metods tat extend te ideas in [9] to te case of convection can be found in recent works [0, 6]. Tese metods are constructed by defining a independent flux variable on te element
4 L. DEMKOWICZ AND J. GOPALAKRISHNAN interfaces wic can solved for first, after wic te internal variables can be locally solved for. Wile tis can be tougt of as akin to static condensation, additional advantages can be exploited, suc as easy stabilization [6] using a penalty parameter. However, p- independent stability for suc metods as not been proved yet. Wile we borrow te idea of letting te fluxes be independent variables in te design of our metod, our metod does not ave stabilization parameters, and as p-independent stability. We organize our presentation suc tat a spectral version of te metod is first exibited (in Section 2). Details regarding te new space of test functions and te stability estimates for te metod on a single element are presented in tat section. Section 3 ten presents te composite metod on a triangular mes. Optimal L 2 error estimates are proved in Teorem 3.2 tere. We conclude in Section 5 opining on important future directions. Proofs of a few tecnical estimates are gatered in Appendix A. 2. Te spectral metod on one element We start by considering te one-element case to fix te ideas and study te element spaces. In oter words, we let Ω be an N-simplex (N = 2, 3 are te cases of most practical importance) wic we denote by K trougout tis section. We now describe and analyze te DPG metod on te single element K. Define te outflow and inflow boundaries of K by out K = { x K : β n > 0}, in K = { x K : β n < 0}. Here, and trougout te paper, we use n to generically denote te unit outward normal on te boundary of a polygonal domain (te domain will vary at different occurrences of n, but will always be clear from te context, e.g., above it is K, wile in (2) it was Ω). Note tat we omitted te β n = 0 case in (9). Tis, and any concerns regarding computational inaccuracies in determination of strict inequalities of (9), will be allayed in 2.3. 2.. Petrov-Galerkin metod using a new test space. Before we describe te metod, we need to introduce a new test space. For tis, we need a coordinate system aligned wit te flow see Figure. Let e (),..., e (N), denote te standard unit vectors and let e β denote te unit vector in te e β -direction. Completing { e β } to form an ortonormal basis { e β, e (2) (N) β,... e β } for R N, we write te new coordinates as η i, i.e., x = x e () + + x N e (N) = η e β + η 2 e (2) β + + η N e (N) β. Any function defined on out K can be extended into K in suc a way tat te extension is constant along te β-direction, i.e., te extension is independent of η. We call tis te extension from outflow boundary and denote it by E out. We can tink of te outflow surface as te grap of a function η = F out (η 2,..., η N ). Ten, a function on te outflow surface can be expressed (after eliminating te η variable) as φ(η 2,..., η N ). Tese coordinates make te expression for E out φ trivial, namely E out φ(η, η 2,..., η N ) = φ(η 2,..., η N ). Let us consider te question of computing a polynomial approximation of te transport solution u satisfying () wit Ω = K. Te starting point in deriving te DPG metod is (9a) (9b) (0)
DPG METHOD 5 η = (u, β v) K + β n u, v K 2.. Petrov-Galerkin metod using a new test space Note tat if out = (u, β v) K + φ, v outk + β n g, v in K () V (K) consists of wit φ = β n u on Before out K, we tedescribe outflux variable. te metod, we need to introduce Te metod finds an approximation to te pair (u, φ) using te spaces non-polynomial sp a new test space. For tis, we need a coordinate system M l ( out K) = {µ : µ F P l (F ), for all faces F witin K. aligned wit te flow. oflet K contained e (),...,e in (N) out, K}, denote te standard unit vectors V l (K) = and η P l let (K) e + β (2a) Te spectral DP Edenote out (M l+ te ( out unit K)). vector in te (2b) approximations (u Here, for any domain e β -direction. D, we use Completing P l (D) to denote {e β } to teform spacean of polynomials ortonormal of degree at most l, and fying basis {e β,e (2) β η,...e(n) β } for R N P l (K) = {η p l : p l, we P l write (K)}. te new coordinates as η i,i.e., (u p, β v Note tat if out K consists of a single face of K, ten V l (K) consists of polynomials, but in general, V l (K) is a non-polynomial space, as it can ave lines of discontinuity witin K. Te spectral DPG metod, motivated x = x e () by (), defines + + x N e (N) approximations (u p, φ p+ ) P p (K) M p+ ( out K), satisfying (0) for all v V = η e β η 2 e (2) β + + η N e (N) p (K). (u p, β v) K + φ p+, v outk β. = (f, v) K (3) β n g, v in K, a((u p,φ p+ ),v) Any function defined on out can be extended into K for all v V p (K). For compact notation, we write in suc a way tat te extension is constant along te β- b(v) a( (u p, φ p+ ), v) def = (u p, β direction, i.e., te extension is v) K + φ p+, v outk independent of η. We call X p (K) b(v) def = (f, v) K β n g, v in K, tis te extension from outflow boundary and denote it by X p (K) def = P p (K) M p+ ( out K), so tat (3) can E out. so tat (3) can be written as te problem of finding (u p, φ p+ ) X p (K) suc tat (u p,φ p+ ) X p (K Let us consider te question of computing a polynomial a( (u p, φ p+ ), v) = b(v) v V p (K). approximation of te transport solution u satisfying () a((u p,φ p wit Ω = K. Te starting point in deriving te DPG Tis is a Petrovtrial spaces are metod is to multiply (a) by a test function v and inte- di using te spaces η 2 β K M ( out K)= V (K) = Figure. Streamline coordinates regarding computational inaccuracies in determination of to multiply (a) strict by a inequalities test functionof v and (9), integrate will be allayed by parts: in 2.3. (f, v) = ( β u, v) K Here, for any dom of polynomials of η P (
6 L. DEMKOWICZ AND J. GOPALAKRISHNAN Tis is a Petrov-Galerkin metod because te test and trial spaces are different. Te solvability of tis metod will be clear sortly, as soon as we study te new finite element space. Proposition 2.. Te component spaces η P p (K) and E out (M p+ ( out K)) are linearly independent, so tat V p (K) can be written as te direct sum V p (K) = η P p (K) E out (M p+ ( out K)). (4) Proof. Suppose tere is an r P p (K) and µ M p+ ( out K) suc tat η r = E out µ, ten, letting η denote te distributional derivative wit respect to te variable η, we ave η (η r) = η (E out µ) = 0. Integrating from te η = 0 plane (extending te polynomial r to all R 3 if necessary) we find tat t r(t, η 2,..., η N ) = t 0 η (η r(η,..., η N )) ds = 0, for any t, so r 0, wic in turn implies tat µ = 0 as well. Tus te sum in (4) is indeed a direct sum. Te degrees of freedom of V p (K) can be inferred from te next lemma. Lemma 2.. Given any µ in M p+ ( out K) and any w in P p (K), tere is a unique v in V p (K) satisfying β v = w, (5a) v outk = µ. (5b) Proof. We construct te required v as a sum of two functions v and v 2, defined below. First set η v (η,..., η N ) = 0 β w(s, η 2,..., η N ) ds, were β denotes te lengt of te vector β. Note tat v 2, wen restricted to eac face of out K is a polynomial of degree at most p +. Now let v 2 = E out (µ v outk). It is easy to verify tat v = v + v 2 satisfies bot te equalities of (5). Te uniqueness of v is also easy to see: If v satisfies (5) wit w = 0 and µ = 0, ten integrating into K from te outflow boundary along te β-direction, we find tat v 0. One can now define a new finite element formally in te sense of Ciarlet [7] as follows. Let Σ denote te set of linear functionals l q, l η defined by l q (v) = ( β v, q) K, for all q P p (K), l η (v) = v, η F, for all η P p+ (F ), for all faces F out K. Ten, wit Σ as te set of degrees of freedom (d.o.f), te geometry-space-d.o.f triple (K, V p (K), Σ) defines a unisolvent finite element because of Lemma 2.. Altoug V p (K)
DPG METHOD 7 may contain discontinuous functions, implementation of tis finite element can proceed using standard finite element tecnology using te above d.o.fs. Proposition 2.2. Tere is a unique solution to te spectral DPG metod (3). Proof. Te system (3) is square. Tis follows because te space of trial functions as dimension dim(x p (K)) = dim(p p (K)) + dim(m p+ ( out K)) = dim(η P p (K)) + dim(e out (M p+ ( out K))) wic equals te dimension of te test space V p (K), due to Proposition 2.. Hence it suffices to sow tat if te rigt and side of (3) vanises, te only possible solution is trivial. But tis is immediate from Lemma 2., by wic we may coose v suc tat β v = u p and v outk = φ p+. Let us now outline an error analysis. We want to apply Teorem.. Te continuity of te form a(, ) is evident from a( (u p, φ p+ ), v) = (u p, β v) K + φ p+, v Ko u p K β v K + /2 K φ p+ outk /2 K v outk (6) ( u p 2 K + K φ p+ 2 outk) /2 v K were we ave cosen te test space norm ( v K = β /2 v 2 K + K v 2 outk). Te inf-sup condition also olds, because applying Lemma 2., we can find a v suc tat β v = u p and v outk = K φ p+, so a( (u p, φ p+ ), v) sup v V p(k) v K = u p 2 K + K φ p+ 2 outk v K ( up 2 K + K φ p+ 2 outk) /2 v K v K. Tus applying Teorem., we obtain te following error estimate for te spectral metod: u u p 2 K + K φ φ p+ 2 K ( 2 inf u wp 2 K + K φ µ p+ K) 2. (w p,µ p+ ) X p(k) Here K = diam(k). Reviewing te above argument, te introduction of factors K and K in (6) may seem arbitrary, and indeed it is. Wile we cose tese factors above for simplicity, later we will need to introduce different mes dependent factors to obtain stability in more appropriate norms. Altoug we used te inf-sup condition to motivate te design of our metod, te way we used it above does not provide us wit te best error estimate possible. It is possible to obtain better error estimates, and indeed to sow tat te error for eac variable is decoupled. In fact, bot variables possess superconvergence in te spectral case, as we see next.
8 L. DEMKOWICZ AND J. GOPALAKRISHNAN Teorem 2.. Let Π W and Π M denote te L 2 -ortogonal projections into P p (K) and M p+ ( out K), respectively. Ten φ p+ Π M φ = 0, u p Π W u = 0. Proof. Subtracting () from (3), we find tat (u p Π M u, β v) K + φ p+ Π M φ, v outk = 0 for all v in V p (K). By Lemma 2., we can coose v in V p (K) so tat v outk = E out (φ p+ Π M φ) and β v = Π M u u p. Hence te identities of te teorem follow. 2.2. Local stability estimates. Altoug our error analysis of te spectral metod is already completed (by Teorem 2.), we want to obtain a few refined stability estimates for te solutions. Tis will be needed later, wen we analyze te multi-element version of te metod. To tis end, we introduce te following local solution operators: Q p : L 2 ( in K) M p+ ( out K), U p : L 2 ( in K) P p (K). Given φ in L 2 ( in K), te local solutions Q p φ and U p φ are defined by te spectral metod (3) wit f = 0, element by element, as follows: (U p φ, β v) K + Q p φ, v outk = φ, v in K (7) for all v V p (K). Also define solution operators for element sources by Q f p : L 2 (K) M p+ ( out K), U f p : L 2 (K) P p (K) (U f pz, β v) K + Q f pz, v outk = (z, v) K, (8) for all v V p (K) and any z in L 2 (K). Te next teorem is te main result of tis subsection. Let β,k denote te lengt of te longest line segment, in te β-direction, contained in K, wit one endpoint in in K and te oter in out K (as marked in Fig. 0(b)). We denote by n F a unit normal to a mes face F. For any subcollection F of te set of faces of K were β n 0, we define te following norms: Here te sums run over all mes faces F contained in F. ψ 2 β,f = F F β n F ψ 2 F, (9) v 2 β,f = F F β n F v 2 F. (20) Teorem 2.2. Te local solution operators ave te following stability estimates: Q p φ 2 β, outk φ 2 β, ink, U p φ 2 K β,k β φ 2 β, ink, (2a) (2b)
DPG METHOD 9 8 L. DEMKOWICZ AND J. GOPALAKRISHNAN outk ( β n 0ere) K outk β 0 β Figure 2.. AAoutflowedgedegeneratingas degenerating as β β 00 and 2.3. Remarks on te robustness of te outflux variable. Notice tat on faces were β n =0,weavenotyetdefinedanyoutfluxapproximation. Q f Sinceteexactoutflux pz 2 (defined in (26)) vanises on suc faces, we β,k β, outk set β te z 2 K, (22a) discrete outflux φ to zero on tose faces. Computationally, te identification Uof f pz inflow 2 K 2 β,k and outflow faces can only be as accurate as te round-off errors permit, i.e., te strict inequalities β 2 z 2 K, (22b) (9)canonlybeimplementedup for to round-off. all φ L 2 ( Tis begs te question: On faces F were β in K) and z in L 2 (K). n F is close to zero, is φ p too sensitive to weter we set F to be an inflow or outflow face? AWe proof nowof argue te teorem tat it is is not. givenfor in Appendix clarity, consider A. te situation of Figure (wic is entirely typical). Te solution φ p of (3) can be written as φ p = Q p g + Q f pf. Ten,byte 2.3. estimates Remarks of Teorem on te2.2, robustness we ave of te outflux variable. Notice tat on faces were β n = 0, we ave not yet defined any outflux approximation. Since te exact outflux (defined in (26)) vanises on suc faces, we set te discrete outflux φ φ p β, outk g β, ink + /2 to zero on tose β,k faces. β f K. Computationally, te identification of inflow and outflow /2 faces can only be as accurate as te round-off errors permit, Tis continues to old as β i.e., te strict inequalities (9) can only be implemented up to round-off. Tis begs te question: n approaces On zero faceson F an were outflow β n face F is(as close in Fig. to zero, ). Note is φ p tat too sensitive te rigtto and weter side remains we set Fbounded to during tis limiting process. But since te left and side norm contains te factor β be an inflow or outflow face? We now argue tat it is not. For n /2 clarity, on te consider outflowteface situation were β of Figure n approaces 2 (wic zero, is entirely te solution typical). φ p must Teapproac solution φzero p of tere. (3) can be written as φ p = Q p g + Q f pf. Ten, by te estimates It is tis of robustness Teorem 2.2, in we φ p ave tat led us to introduce it as te new outflux variable to be solved for. Te coice of suc a variable is not traditional. In ybridized metods for elliptic problems [5] one usually sets te trace of te primal variable u as a new unknown. φ In tis spirit, one could contemplate p β, outk a metod g β, ink were + /2 β,k β one f finds K. /2 u p and λ p satisfying Tis continues to (uold p, β as β v) n K approaces + β n λ p,v zero outk on=(f,v) an outflow K face β n (as g, v in in Fig. K, 2). Note tat (23) te rigt and side remains bounded during tis limiting process. But since te left and side instead normof contains (3). Suc te metods factor ave β n appeared /2 on te in te outflow literature face were testandard β n approaces UDGmetod zero, te cansolution be recast φ p into must ybrid approac formzero wittere. upwind traces set as a new unknown λ,seee.g., te Itmetod is tis robustness in [6] for te in purely φ p tat convective led us tocase introduce and teitrecent as teworkof[]).however,an new outflux variable to be equation solved suc for. as Te (23) coice can lead of suc to matrices a variable close is not to singularity traditional. inin situations ybridized likemetods in Figurefor. elliptic Indeed, problems te λ in [9] (23) onecannot usuallybe sets uniquely te trace defined of te were primal β n variable =0. u as a new unknown.
0 L. DEMKOWICZ AND J. GOPALAKRISHNAN In tis spirit, one could contemplate a metod were one finds u p and λ p satisfying (u p, β v) K + β n λ p, v outk = (f, v) K β n g, v in K, (23) instead of (3). Suc metods ave appeared in te literature te standard UDG metod can be recast into ybrid form wit te upwind traces set as a new unknown λ, see e.g., te metod in [0] for te purely convective case). However, an equation suc as (23) can lead to matrices close to singularity in situations like in Figure 2. Indeed, te λ in (23) cannot be uniquely defined were β n = 0. 3. Approximations using a triangular mes In tis section, we move to te multi-element case, restricting ourselves to te case of two space dimensions. We consider te composite DPG metod on a mes of te domain Ω, and a general polynomial degree p. Since te degree is uniformly set to p on all elements tis is te p-version of te metod. We assume tat te domain Ω is mesed by a geometrically conforming mes of triangles, denoted by T, and set = max{ K : K T }. Define te (discontinuous) finite element spaces W = {w : w K P p (K), for all mes elements K T }, M = { µ : µ F P p+ (F ), for all mes edges F not on in Ω}, V = { v : v K V p (K), for all mes elements K T }. Here V p (K) is te new finite element we introduced in Section 2. (24a) (24b) (24c) 3.. Definition of te composite metod. We put togeter te spectral metod on eac element to get te following composite metod on te mes T : Find (u, φ ) def = W M satisfying X for all v V, were a ( (u, φ ), v ) = (f, v ) Ω β n g, v in Ω, (25) a ( (u, φ ), v ) = ] [ φ, v outk φ, v ink\ inω (u, β v ) K. K T We call φ te outflux approximation, because on comparison wit (), we expect it to approximate φ def = βu n + (26) on every mes edge F, were n + is te unit normal on F wit its direction cosen so tat β n + > 0. On edges were β n = 0, we set φ = 0. Equation () also sows tat tis φ togeter wit te exact solution u, satisfies a ( (u, φ), v) = (f, v) Ω β n g, v in Ω, (27)
DPG METHOD for all v in V = {ν L 2 (Ω) : ν K V (K) for all mes elements K}, were V (K) is as in (55). In oter words, te DPG metod is consistent. In order to simplify notation and write φ, v in K for φ, v in K\ in Ω, we identify te space M wit its extension by zero to in Ω, i.e., we identify M and {µ : µ F P p+ (F ) for all mes edge F and µ in Ω = 0}, to be te same (cf. te definition in (24b)). 3.2. Mes dependent norms. We collect te definitions of various mes dependent norms ere for ready reference. First, we extend te definition of te norms ψ 2 and v β,f β,f to more general collections F of edges from te entire mes (now not necessarily edges of one triangle) simply by letting te sums in (9) and (20) runs over all suc edges in F. As before, only edges wit β n 0 are picked. Using tese extended definitions, we now define more norms on a subcollection of mes elements R, as follows: v 2,β,R = β,k v 2 β,, (28) outk K R ψ 2 =, β,r β,k ψ 2 β K R ψ 2 = ψ 2, β,r β,k β K R, outk, (29), outk, (30) ψ 2, β,r = ψ Q pψ 2, β,r. (3) Tese norm notations ave mnemonic subscripts of, /, β, or /β etc., wic indicates a factor of β,k, / β,k, β n, or / β n (resp.) in te sums over mes objects defining te norms. Wit tese notations, te norm on te trial space is defined by (u, φ) 2 = K T ( u U p φ 2 K + ψ Q p ψ 2, β,k Te norm on te test space is defined by v 2 = K T β v 2 K + v 2,β,K. It is obvious from te construction of V p (K) tat te above is a norm on V. To sow tat (u, φ) is a norm on te test space, we first need to recall te following. Lemma 3.. In any triangulation T of Ω, tere is at least one triangle K T suc tat in K in Ω. Proof. See [5]. Proposition 3.. Te functional (u, φ) is a norm on te space X = L 2 (Ω) M, were M = {µ : µ F L 2 (F ) for all mes edges F and µ in Ω = 0}. Proof. Suppose (u, φ) = 0. Ten, in particular φ Q p φ β, outk = 0 (32) ).
2 L. DEMKOWICZ AND J. GOPALAKRISHNAN for all elements K. By Lemma 3., tere is an element K wit in K in Ω. On tis element, φ in K = 0, so Q p φ outk = 0. Hence (32) implies tat φ outk = 0. Proceeding inductively over all elements we find tat φ = 0. Tis in turn implies tat u U p φ = u = 0. 3.3. Preliminary error analysis. It is possible to get stability and error estimates for te DPG metod in te mes dependent norm immediately. Solvability of te metod follows as a particular consequence. Teorem 3.. Te bilinear form a (, ) satisfies te inf-sup and continuity conditions: olds for all u W and φ M, and a ( (u, φ ), v ) sup (u, φ ), v V v a ( (u, φ), v ) (u, φ) v for all u L 2 (Ω), φ M, and v V. If (u, φ) and (u, φ ) are te exact and discrete solutions, ten (u u, φ φ ) 2 2 inf (u w, φ µ ) 2. (w,µ ) W M Proof. We first rewrite (25) using te definition of te local solution operators in (7) as follows: a ( (u, φ ), v ) = K T [ (u, β v ) K + φ, v outk] φ, v in K = K T (u U p φ, β v ) K + φ Q p φ, v outk. (33) Let us prove te inf-sup condition. By Lemma 2., we can coose v element by element, suc tat β v K = u U p (φ in K ), Ten ( φ v outk = Q p (φ in )) K outk β,k β. n a ( (u, φ ), v ) = K T u U p φ 2 K + φ Q p φ 2, β,k = (u, φ ) v. Tis proves te inf-sup condition. Te proof of te continuity inequality also follows from (33) after applying te Caucy- Scwarz inequality. Te error estimate ten immediately follows from Teorem..
DPG METHOD 3 2 L. DEMKOWICZ AND J. GOPALAKRISHNAN S 3 S 2 S S 0 S 2 S S 0 S S 0 S 0 Figure Figure 2. Amesbeingsplitintolayers(flowisalongtepositivey-direction). 3. A split into layers is along te positive y- direction). By Lemma 3., S 0 is not empty. Next, setting T (0) = T,werecursivelydefine,forall integers l>0, te sets Te above analysis is in te most natural norms suggested by te metod itself. In fact, we ave sown above T (l) tat = T (l ) \ S l S l = {K a T (l) : in K in Ω (l) }, ( (u, φ ), v ) sup = (u, φ ) were Ω (l) is te domain formed by te union of all triangles of T (l), v.repeatedapplication V v of Lemma 3. sows tat S l is nonempty for every l, unlesst (l) is empty. So tis process exausts for all allu elements W, of φ te Mmes,. Noneteless, at some finite convergence value ofrates l, say inl standard = n. Te norms domain (like L 2 ) def are not obvious Λ l = Ω\ Ω (l) from te teorem. Te remainder of tis section is devoted to obtaining is mesed by S 0 S S l.itsoutflowboundaryisdenotedby convergence rates in L 2 -like norms. Γ l = out Λ l. All mes 3.4. Aedges, Poincaré excluding inequality. tose on in As Ω, aand first excluding step to tose obtainwere stability β n =0,arecontained and error estimates in in Γ l Lfor 2 -norm, some lwe prove n. Wittesenotationsandobservations,weimposeamildassumption a Poincaré inequality. Notice tat te norm in (3) is applied to te on te difference mes and ψ prove Q te Poincaré inequality. outk p(ψ in ). Tis is te difference between te outflow values of ψ and its K Assumption inflow values 3.. Define transported te layer ontowidt out K. bydiscrete analogues of te Poincaré inequality bound L 2 -like norms using sum of squares of differences (under some discrete analogue of a zero boundary condition). Terefore, d l =max{ it is peraps β,k : K not S surprising l }. tat suc a discrete Poincaré We assume type estimate tat our can meses also be arefound suc for tate norm in (3) (recalling tat any ψ in M satisfies a zero boundary condition on te inflow n boundary). To establis te inequality, we first dneed l Lto Ω define layers of mes elements marcing from in Ω. Te first layer is definedl=0 by were L Ω is a fixed constant depending on Ω, but independent of te meses and element sizes. We also assume tat tere iss a 0 fixed = {Kconstant T : C in K suc in tat Ω}. if K and K are any two
4 L. DEMKOWICZ AND J. GOPALAKRISHNAN By Lemma 3., S 0 is not empty. Next, setting T (0) integers l > 0, te sets Λ l T (l) = T (l ) \ S l S l = {K T (l) : in K in Ω (l) }, = T, we recursively define, for all were Ω (l) is te domain formed by te union of all triangles of T (l). Repeated application of Lemma 3. sows tat S l is nonempty for every l, unless T (l) is empty. So tis process exausts all elements of te mes, at some finite value of l, say l = n. Te domain def = Ω \ Ω (l) is mesed by S 0 S S l. Its outflow boundary is denoted by Γ l = out Λ l. All mes edges, excluding tose on in Ω, and excluding tose were β n = 0, are contained in Γ l for some l n. Wit tese notations and observations, we impose a mild assumption on te mes and prove te Poincaré inequality. Assumption 3.. Define te layer widt by d l = max{ β,k : K S l }. We assume tat our meses are suc tat n d l L Ω l=0 were L Ω is a fixed constant depending on Ω, but independent of te meses and element sizes. We also assume tat tere is a fixed constant C suc tat if K and K are any two neigboring elements, ten eiter te inequality olds or it olds after excanging K and K. β,k C β,k Above and in te remainder, te letter C, wit or witout subscripts, denotes generic constants independent of elements K and polynomial degree p. Teir value at different occurrences may vary. Assumption 3. permits quite general meses, including anisotropic refinements (e.g., to capture sock waves). Loosely speaking, te first part assumes tat elements witin eac layer ave about te same lengt in te flow direction. Te second part assumes tat element lengts in te flow direction for neigboring elements are comparable. Quasiuniform meses obviously satisfy te assumption. Teorem 3.2. If Assumption 3. olds, ten for all ψ M, we ave ψ 2, β,t C 0L 2 Ω ψ 2, β,t. (were C 0 is independent of T and p, and te norms are as in (29) and (3)). Proof. Te first step is a local inequality. On any triangle K, ψ β, outk ψ Q pψ β, outk + Q pψ β, outk ψ Q p ψ β, outk + ψ β, ink,
DPG METHOD 5 were te last inequality was due to Teorem 2.2 (2). Consequently, ψ 2 β, outk ψ 2 β, ink ( ψ β, outk + ψ ) (34) β, ink ψ Qp ψ β, outk olds on every mes element K. Te next step is to sum over R l = S 0 S S l. Ten, we ave ) K R l ( ψ 2 β, outk ψ 2 β, ink K R l ( ψ β, outk + ψ β, ink) ψ Qp ψ β, outk. Wen rewriting te left and side as a sum over te mes edges, we observe tat all contributions due to edges interior to Λ l cancel out. Furtermore, since ψ M, it vanises on te inflow boundary in Λ l in Ω. Terefore only contributions on te outflow boundary of Λ l, namely Γ l, remains, i.e., ψ 2 (35) β,γ l ( ψ β, outk + ψ ) β, ink ψ Qp ψ, outk. β K R l We apply Caucy-Scwarz inequality to te rigt and side of (35) to get ( ψ 2 ( 2 β,γ l β,k ψ 2 + ) ) /2 β, outk ψ 2 β, ink K R l ( ) /2 ψ Q p ψ 2 β,k β, outk K R l By Assumption 3., te lengts β,k for neigboring elements are comparable, so tat wic implies K R l β,k ψ 2 β, ink C K R l β,k ψ 2 β, outk ( ) /2 ψ 2 C β,γ l β,k ψ 2 β, outk K R l ( ) /2 ψ Q p ψ 2 β,k β, outk K R l l=0 = C ψ, β,r ψ l, β,r. l Te final step involves multiplying by te layer widt d l, and summing over l, n n d l ψ 2 d β,γ l l C ψ, β,r ψ l, β,r l l=0 CL Ω ψ, β,t ψ, β,t, (36)
6 L. DEMKOWICZ AND J. GOPALAKRISHNAN were we increased te norms over R l to norms over te wole mes T. By te definition of d l, we know tat so we ave β,k d l, for all K S l, ψ 2, β,t = n l=0 β,k ψ 2 β, outk K S l n d l ψ 2 β, outk l=0 K S l L Ω C ψ, β,t ψ, β,t, were te last inequality is due to (36). Tis proves te result. 3.5. L 2 -error estimates. Now we prove an optimal error estimate for u in te standard L 2 (Ω)-norm, as well as error estimates for te outflux approximation φ in L 2 -like norms on mes edges. Te estimates are wit constants independent of te polynomial degree p and mes T. Teorem 3.3. Suppose (u, φ) is te exact solution, (u, φ ) is te discrete solution. Ten we ave te following error estimates: If Assumption 3. olds, letting we ave φ φ 2 3 inf φ ψ, β,k p+ 2 ψ p+ M β,k β ( ) ɛ(φ) def = inf φ ψ 2 + C ψ M, 0L 2 β,t Ω φ ψ 2,, β,t, K. (37) φ φ 2 2 ɛ(φ) and (38), β,t u u 2 Ω 4 β ɛ(φ) + 2 inf u w 2 w W Ω. (39) Proof. Te first step of tis proof consists of identifying te equations satisfied by te following discrete error functions: ε u = Π W u u, ε φ = Π Mφ φ,
DPG METHOD 7 were Π W and Π M are L 2 -ortogonal projections into W and M, respectively. Subtracting te exact and discrete equations, 0 = K T ( (u u, β v ) K ) + φ φ, v outk φ φ, v in K = K T ( (Π W u u, β v ) K + Π M φ φ, v outk φ φ, v in K were te introduction of Π W is possible as β maps V p (K) into P p (K). Notice tat in te last term we may not replace φ by Π M φ as te inflow traces of v need not be polynomial. But we may add and subtract Π M φ witin it. Ten, we can express te identity using te discrete error functions as follows: or equivalently (see (33)), a ( (ε u, ε φ ), v ) = K T φ Π M φ, v in K, ), on every mes element K. In oter words, (ε u U p ε φ, β v ) K + ε φ Q pε φ, v outk = φ Π M φ, v in K, ε u U p ε φ = U p(φ Π M φ), ε φ Q pε φ = Q p(φ Π M φ). We sall use tese identities to bound te errors. Te second step of tis proof involves bounding ε φ. By (4b), (40) (4a) (4b) ε φ Q pε φ β, outk = Q p(φ Π M φ) β, outk φ Π M φ β, ink, (42) by te bound (2) on Q p ( ) of Teorem 2.2. Terefore, φ φ, β,k φ Π M φ, β,k + εφ, β,k = /2 β,k (φ Π Mφ) Q p (φ Π M φ) β, outk + /2 β,k εφ Q pε φ β, outk /2 β,k φ Π Mφ β, outk + 2 /2 β,k φ Π Mφ β, ink. By te inequality of aritmetic and geometric means, tis implies φ φ 2, 3,K β,k φ Π Mφ 2, β β, outk ink
8 L. DEMKOWICZ AND J. GOPALAKRISHNAN wic gives te first error estimate of te teorem. Te tird step again involves a bound on ε φ, but now in a weaker norm. We use te Poincaré inequality of Teorem 3.2: ε φ 2 C, 0L 2 β,t Ω ε φ 2, β,t C 0 L 2 Ω φ Π M φ 2 β, in K, (43) β,k K T were we ave also used (42) in te last step. Splitting φ φ = (φ Π M φ) + ε φ as before, and estimating, we prove te second estimate (38) of te teorem. Now we come to te final step, were we obtain te error estimate for u, namely (39), as follows: ε u K U p ε φ K ε u U p ε φ K by triangle inequality = U p (φ Π M φ) K by (4a) /2 β,k β /2 φ Π Mφ β, ink by Teorem 2.2. Using Teorem 2.2 again, we ave ε u K /2 β,k ( ε φ β /2 β, ink + φ Π ) Mφ β, ink. Summing over all elements and using (43), we obtain ε u 2 Ω 2 ( ε φ β 2 +, β,t β,k φ Π M φ 2 ) β, in K K T 2 β K T ( C0 L 2 Ω β,k φ Π M φ 2 β, in K + β,k φ Π M φ 2 β, in K ), from wic te estimate (39) follows. 3.6. Interior fluxes by postprocessing. Te advective flux is q = βu. A natural approximation to tis is q = βu, were u is te computed solution. However q is not conservative. It is possible to easily adapt an idea of [8] to generate a conservative flux by a simple postprocessing sceme. Our postprocessed flux is denoted by q. Its restriction to eac element K lies in te Raviart-Tomas space P p+ (K) + xp p+ (K) and is defined by ( q, r) K = ( β u, r) K, r P p (K), (44a) q n, µ F = ϕ, µ n F, µ P p+ (F ), (44b)
DPG METHOD 9 were (44b) is imposed on all edges F of K, φ n, on out K, φ n, on in K \ in Ω, ϕ = β g, on in K in Ω, 0, on faces were β n = 0, and, as before, n denotes te unit outward normal of K. Since (44) uses te well known degrees of freedom of te Raviart-Tomas space, it is easy to see tat (44) uniquely defines a q in H(div, Ω). Teorem 3.4. Write u β def = β u = q and u β, def = q. Ten, q is conservative, and u β, Π p+ u β = 0 (45) were Π p+ is te L 2 (K)-ortogonal projection on P p+ (K). Proof. First of all, observe tat P p+ (K) V p (K). Indeed, if z p+ is any polynomial in P p+ (K), ten finding a v in V p (K) as in Lemma 2. suc tat β v = β z p+, (46a) v outk = z p+ outk, (46b) we find from (46a) tat z p+ v must be a function tat is constant along te β-direction. But te same function vanises on out K by (46b), so it must vanis everywere. Hence z p+ v V p (K). As a consequence, we ave, for all z in P p+ (K), ( q, z) K + q n, z K = (u, β z) K + φ, z outk φ, z in K + β n g, z in K in Ω by (44), = (f, z) K = ( β u, z) K by (25). Wit q = βu, we can rewrite tis as ( q, z) K = ( q, z), for all z P p+ (K), (47) wic proves (45). To sow tat q is conservative, let D be any subdomain formed by a union of some or all of te mes elements. Ten (47) applied wit z = implies q n ds = q n ds, D were we ave also used te H(div, Ω)-conformity of q. Tis sows tat te net outward fluxes of q and q coincide on any suc subdomain D. D 4. Numerical studies We present a few small scale numerical studies to illustrate and confirm te teory. For more realistic and larger size simulations, we will need full p adaptivity, wic will be presented elsewere.
20 L. DEMKOWICZ AND J. GOPALAKRISHNAN 4.. Implementation aspects. In comparison wit te UDG metod, we do ave more equations to solve. However, te reason our metod remains competitive is tat we are able to solve for te fluxes (φ ) first and locally recover u afterwards. Tis results in a dimensional reduction tat is quite attractive for ig p. Indeed, wit respect to p, te number of degrees of freedom for φ is O(p), wile te corresponding number for u is O(p 2 ). Tis dimensional reduction is quite analogous to ybridized metods [9], were te so-called Lagrange multipliers are solved for first, and te remaining variables are recovered locally, element by element. To state te matrix form of te metod, we need bases for W, M, and V. For M, we set a basis consisting of functions supported only on one edge, wile te bases for te oter spaces consist of functions supported on only one triangle. Let Φ and U denote te vector of coefficients of φ and u in teir basis expansion, respectively. To sow ow one solves for fluxes first, observe tat te global finite element space V can be split into te outflow extensions E out (M ) plus a linearly independent remainder R. We enumerate te global basis for V suc tat te basis functions in E out (M ) come first. Wit v set to suc test functions, (25) reduces to an equation involving only φ : [ ] φ, v outk φ, v in K\ in Ω (48) K T = (f, v ) Ω β n g, v in Ω for all v E out (M ). As a consequence, te stiffness matrix of (25) is block triangular: ( ( ) A 0 Φ = F (49) B C) U were A, B and C are sparse matrix blocks, and F is te corresponding load vector. Tis clearly sows tat we can solve for Φ first and te result can ten be used to compute u (or U) locally, element by element. Additionally, note tat in te decomposition V = E out (M ) R, all te functions in V aving discontinuities inside elements lie in te E out (M )-component. Terefore, wen making te stiffness matrix of te bilinear form a ( (u, φ ), v ), tere is no need to integrate basis functions wit discontinuities witin te element. Indeed, as seen in (48), te term involving integration witin elements, namely (u, β v ), vanises for v in E out (M ), so tere is no need to compute tis integral. Finally, suppose we enumerate te basis for M in suc a way tat functions on Γ l appear before tose on Γ l (see te definition of te layers in 3.4). Ten te matrix A in (49) is a square triangular matrix. In tis case, Φ can be found fast using backsubstitution. Since tis is te only globally coupled system to solve, te remaining (local) computations for u can be locally optimized (or parallelized). 4.2. A one-dimensional example. We begin wit a simple one-dimensional example to sow te enanced accuracy and stability of te DPG metod compared to te DG metod. Te model problem on Ω = (0, ) tat we sall consider is u = f, u(0) = u 0, wit data u 0 R and f L 2 (0, ) set so tat te exact solution is u(x) = arctan(α(x x 0 ))
DPG METHOD 2 0 0 DPG DG L 2 error in u 0 0 2 0 2 4 6 8 p Figure 4. Convergence for te one-dimensional example were x 0 = and α = 00. We use te analogue of our spectral DPG metod (of Section 2) in one dimension, i.e., K = (0, ) and te test space is V p (K) = P p+ (K). Figure 4 sows te error in u vs. p in a semilog plot. Wile bot metods exibit exponential convergence, te error for te DPG metod is one order less tan tat for te DG metod. Te gap between te two convergence curves grows wit increasing coefficient α tat controls te steepness of te solution. Te different beavior of te two metods is furter illustrated in Figures 5(a) and 5(b) tat sow te graps of te approximate solutions for some values of p. Bot metods deliver te exact value of te flux at x =. Contrary to te DG metod, owever, te DPG solution u witin te element is disconnected from value at and delivers te best approximation error in L 2 -norm. Tis manifests itself wit a less oscillatory beavior of te solution. Our numerics also confirmed Teorem 2., i.e., wen we compared te DPG solution wit te L 2 -projection of te exact solution, we found te difference to be zero (of te order of round-off errors). 4.3. and p convergence to a solution wit discontinuity. Following [2], we consider a square domain Ω = (, ) 2 werein an advection problem is set up so tat β = (, 9/0) and te exact solution is sin(π(x + ) 2 /4 sin(π(y 9x/0)/2) for x, 9x/0x < y <, u(x, y) = e 5(x2 +(y 9x/0) 2 ) for x, y < 9x/0. Tis u as a discontinuity along te line y = 9x/0. As in [2], we first mes Ω by an n n quadrilateral mes aligned wit te line of discontinuity (see [2, Fig. 0] for details). Since our metod is based on triangular meses ([2] only considered quadrilateral meses), we furter split eac of te quadrilaterals into two triangles by connecting teir diagonals of positive slope. Anoter difference wit te experiment in [2] is tat wile [2] needed to include a stabilizing lower order reaction term for teoretical reasons, we do not need to,
22 L. DEMKOWICZ AND J. GOPALAKRISHNAN 0 Te spectral DPG solutions 0.5 u(x).5 2 Exact solution 2.5 p= p=3 p=8 3 0 0.2 0.4 0.6 0.8 x 0 (a) Te DPG solution Te spectral DG solutions 0.5 u(x).5 2 Exact solution 2.5 p= p=3 p=8 3 0 0.2 0.4 0.6 0.8 x (b) Te DG solution Figure 5. Te enanced stability of DPG approximations, compared to te DG ones, in te case of a model transport problem wit a layer at te rigt end ence we solve te purely advective problem. We report te errors in u and φ for te cases n = 5, 9, 7, 33 and p =,..., 5 in Figure 6. It is clear from Figure 6 tat te DPG metod converged under and p refinements as if te solution were infinitely regular (see Figure 6). We observed a similar beavior for te DG metod as well. Suc observations demonstrate te advantage of using discontinuous finite elements wit meses aligned wit te solution discontinuity. For te standard DG metod, tis observation was made in [2], were DG is compared wit te streamline diffusion metod. Wile te streamline diffusion solution can also be improved by aligning meses wit sock lines [2], its convergence rate remains limited by te regularity of te
DPG METHOD 23 (a) Log-log plots of u u for various and p values (b) Log-log plots of flux errors φ φ β,e Figure 6. Te p convergence of te DPG metod (log-log plots) for te Houston-Scwab-Süli example [2] solution, as observed in [2]. Since our purpose is a comparison of DG and DPG, we note ere tat te errors we observed for te DG metod (not sown in te figure), as in 4.2, remained iger tan tat of te DPG metod. Figure 6 suggests tat exponential rates of convergence can be obtained wit te DPG metod, even for discontinuous solutions, once an adaptive strategy to align te meses wit te socks is implemented. Te data tat was used to plot Figure 6 also sows te rate of te convergence. Figure 6(a) indicates tat u u converges at te rate p+ (for p =... 5). Tis is in accordance wit Teorem 3.3. On te oter and, Figure 6(b) sows tat te fluxes converge suc tat φ φ, β,t goes to zero at te rate of p+2.5, wic is one order more tan wat we were able to prove in Teorem 3.3.
24 L. DEMKOWICZ AND J. GOPALAKRISHNAN (a) -refinement: DPG converges at a iger rate (b) p-refinement: DPG gives lower error Figure 7. Te DPG metod performs better tan te DG metod under bot and p refinements for te Peterson example [7]. Bot plots are in a log-log scale. 4.4. Te Peterson example. Next, we consider te well-known Peterson example [7]. Peterson s mes of te unit square is a specific quasiuniform mes of te type appearing in Figure 8, obtained from an n n partition of te unit square, but wit additional vertical lines troug some mes vertices suc tat te new lines divide te unit square into, say m, vertical strips (see [7] for details). On suc meses, we apply te UDG and DPG metod to te problem u/ y = 0, u(x, 0) = sin 6x, x (0, ). Te results in Figure 7 sow tat te DPG metod delivers better results tan te standard DG metod. Figure 7(a) is obtained wit p = and four progressively finer meses: first wit n = 6, m = 3, second
DPG METHOD 25 (a) Te DPG solution (b) Te DG solution Figure 8. DPG solutions exibit less crosswind diffusion tan DG solutions. In tis example, te flow is upward. wit n = 2, m = 6, tird wit n = 24, m = 8, and fourt wit n = 48, m = 6. Tis allows us to study te case of -refinement. On te oter and, Figure 7(b) is obtained by fixing te mes wit n = 6, m = 3, and increasing te degree p from 0 troug 5. Te results clearly demonstrate te well-known 0.5-order suboptimality of te DG metod in te -refinement case, but more interestingly, also sow tat te DPG metod converges at te optimal rate. We also present te solutions u obtained wit bot DG and DPG metods wen te inflow data is u(x, 0) = x 2 in Fig. 8. We used piecewise constant approximations (p = 0). Observe tat as te flow proceeds upwards, te values diffuse orizontally for DG case, wile tere is little indication of suc crosswind diffusion for te DPG metod wit te optimal test functions. 5. Concluding remarks 5.. Summary. We proved optimal error estimates for u in and p. Using te known best approximation estimates [20] for a quasiuniform mes of messize, we obtain as a corollary of Teorem 3.3 tat C u u Ω s p u s H s (Ω) + ( )s2 /2 β /2 p u s H 2 /2 s 2 (Ω) (50) wit some C independent of and p and any 0 s p + and 0 s 2 p + 2. If te solution is smoot, coosing s = p+ and s 2 = p+2, we obtain te optimal -convergence rate of O( p+ ). Te p-convergence rate is also optimal. Note owever tat tere may be room for improvement in te tecniques used in te current proof. In Teorem 3.3 and (50), note tat te regularity on u tat we need to obtain te optimal convergence rate is iger tan expected. Furtermore, for te fluxes, we proved a rate of convergence tat is suboptimal by one order, but numerical experiments