CONSTRUCTIVELY WELL-POSED APPROXIMATION METHODS WITH UNITY INF SUP AND CONTINUITY CONSTANTS FOR PARTIAL DIFFERENTIAL EQUATIONS

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1 MATHEMATICS OF COMPUTATION Volume 82, Number 284, October 203, Pages S (203)02697-X Article electronically publised on April 23, 203 CONSTRUCTIVELY WELL-POSED APPROXIMATION METHODS WITH UNITY INF SUP AND CONTINUITY CONSTANTS FOR PARTIAL DIFFERENTIAL EQUATIONS TAN BUI-THANH, LESZEK DEMKOWICZ, AND OMAR GHATTAS Abstract. Starting from te generalized Lax-Milgram teorem and from te fact tat te approximation error is minimized wen te continuity and inf sup constants are unity, we develop a teory tat provably delivers well-posed approximation metods wit unity continuity and inf sup constants for numerical solution of linear partial differential equations. We demonstrate our single-framework teory on scalar yperbolic equations to constructively derive two different p finite element metods. Te first one coincides wit a least squares discontinuous Galerkin metod, and te oter appears to be new. Bot metods are proven to be trivially well-posed, wit optimal pconvergence rates. Te numerical results sow tat our new discontinuous finite element metod, namely a discontinuous Petrov-Galerkin metod, is more accurate, as optimal convergence rate, and does not seem to ave nonpysical diffusion compared to te upwind discontinuous Galerkin metod.. Introduction Te work in tis paper is inspired from te recent researc on te discontinuous Petrov Galerkin metod (DPG) of Demkowicz and Gopalakrisnan [, 2, 3] for te numerical solution of partial differential equations. Te metod starts by partitioning te domain of interest into nonoverlapping elements. Variational formulations are posed for eac element separately and ten summed up to form a global variational statement. Elemental solutions are connected by introducing ybrid variables (also known as fluxes or traces) tat live on te skeleton of te mes. Tis is terefore a mes-dependent variational approac in wic bot bilinear and linear forms depend on te mes under consideration. In general, te trial and test spaces are not related to eac oter. In te standard Bubnov Galerkin (also known as Galerkin) approac, te trial and test spaces are identical, wile tey differ in a Petrov Galerkin sceme. Traditionally, one cooses eiter Galerkin or Petrov Galerkin approaces, ten proves te consistency and stability in bot infinite and finite dimensional settings. Te DPG metod introduces a new paradigm in wic one selects bot trial and test spaces at te same time to satisfy well-posedness. In particular, one can select trial and test function spaces for wic te continuity and inf sup constants are unity. Given a finite dimensional trial subspace, te finite dimensional test space is constructed in Received by te editor April 8, 20 and, in revised form, October 3, Matematics Subject Classification. Primary 65N30; Secondary 65N2, 65N5, 65N22. Key words and prases. Inf-sup condition, inf-sup constant, generalized Lax-Milgram teorem, discontinuous Galerkin metod, discontinuous Petrov-Galerkin metod, consistency, finite element metod, stability, convergence, well-posedness yperbolic partial differential equations. 923 c 203 American Matematical Society

2 924 TAN BUI-THANH, LESZEK DEMKOWICZ, AND OMAR GHATTAS suc a way tat te well-posedness of te finite dimensional setting is automatically inerited from te infinite dimensional counterpart. Te DPG metod in [2] starts wit a given norm in te trial space and ten seeks a norm in te test space in order to acieve unity continuity and inf sup constants. Anoter DPG metod in [3] acieves te same goal but reverses te process, i.e., it looks for a norm in te trial space corresponding to a given norm in te test space. Clearly, tis is one of te advantages of te DPG metodology, since it allows one to coose a norm of interest to work wit, wile rendering te error optimal, i.e., smallest in tat norm. Furtermore, te DPG metods ave been sown to be robust and provide optimal p-convergence. We sall not discuss te advantages of te DPG metods any furter ere, and te readers are referred to te original DPG papers [, 2, 3] for more details. Here, we pursue a different option. In particular, we sall not prescribe norms in eiter te trial and test spaces. Instead, we permit te structure of te problem to determine its natural energy norms. Our goal is to develop a single framework tat adapts to te problem at and, wile automatically generating accurate finite element metods wit trivially-proven and guaranteed stability. Similar to oter DPG metods, once one cooses te test or trial basis function, te oter as to be solved for. Depending on ow one applies our teory to problems under consideration, basis functions can be trivially obtained or solved for troug an adjoint partial differential equation, as we sall sow. Te remainder of te paper is organized as follows. In Section 2, we first develop a single framework for general variational problems tat can be written in terms of bilinear and linear forms. Ten, we develop a few analytical results tat elp us prove te optimality of te resulting finite dimensional approximations and teir well-posedness. In Sections 3 and 4, we apply our single framework, but using two different points of view, to linear yperbolic equations of advection-reaction type. We sall sow tat te framework constructively leads to an existing stabilized p-discontinuous Galerkin (DG) metod and a new p-dpg metod for advection-reaction equations. We discuss caracteristics of eac metod and teir p-convergence in detail. Te cief purpose of te paper is to introduce our teory and to demonstrate its usefulness to partial differential equations in deriving accurate and stable finite element metods. We furter strengten our findings by several one- and two-dimensional numerical examples in Section 5, and Section 6 concludes te paper. 2. Abstract teory development In tis section, we develop a teory for constructive approximations of linear partial differential equations. Our goal is to construct finite dimensional approximations tat are guaranteed to be trivially well-posed wit unity continuity and inf sup constants. Here, trivial well-posedness means tat te well-posedness of te finite dimensional problems is trivially inerited from teir infinite dimensional counterparts. Te starting point of our teory, to be sown, is not new since our work is inspired by te recent discontinuous Petrov Galerkin (DPG) metodology of Demkowicz and Gopalakrisnan [, 2, 3]. However, we sall point out te differences between our approac and te existing DPG metods.

3 WELL-POSED APPROXIMATION METHODS 925 Let U and V be Hilbert spaces over te real line (generalization of our teory to te complex field is straigtforward). Consider te following variational problem: { Seek u U suc tat () a(u, v) =l(v), v V, were l ( ) is a linear form on V, a(, ) is a bilinear form satisfying te continuity condition wit continuity constant M, (2) a (u, v) M u U v V, te inf sup condition wit te inf sup constant γ, γ >0 : inf sup a (u, v) (3a) γ, u U v V u U v V and te injectivity of te adjoint operator (to be defined), (3b) (a (u, v) =0, u U) (v =0). If (2) and (3) old, ten by te generalized Lax-Milgram teorem [4, 5] (also known as te Banac-Nečas-Babuška teorem [6]), () as a unique solution and te solution is stable in te following sense: u U γ l V, were V is te topological dual of V. Note tat for convenience in writing, we ave abused te notation sup v V instead of sup v V,v 0 (and similarly for inf). Now let U n U and V n V be two finite dimensional trial and test spaces, and consider te following finite dimensional approximation problem: { Seek u U (4) n suc tat a(u,v )=l(v ), v V n. If dim U =dimv = n, and te following discrete inf sup condition a (u,v ) (5) γ > 0 : inf sup γ u U v V u U v V olds, ten te finite dimensional problem (4) is well-posed by an application of te generalized Lax-Milgram teorem for finite dimensional problems (also known as te Babuška s Teorem [4, 7]). In general, owever, te finite dimensional problem (4) does not inerit te well-posedness of te infinite dimensional counterpart () except for some special circumstances. One, terefore, as to prove te nontrivial discrete inf sup condition [6]. In tis paper, we constructively develop a class of finite dimensional approximations in wic te discrete inf sup condition (5) trivially follows from te continuous one (3a). Before doing so, let us first discuss te approximation error between te finite and infinite dimensional solutions. To begin, we recall te following projection result in wic te norms of te projection and its complement are equal. Lemma 2.. Let U U be a subspace of a Hilbert space U. SupposeP : U U, is a projection, i.e., P 2 = P,andP is not null or identity. Ten P = I P, were te norm P is induced from a norm in U, wic is in turn induced from an inner product in U.

4 926 TAN BUI-THANH, LESZEK DEMKOWICZ, AND OMAR GHATTAS Proof. Many proofs of tis result can be found in [8]. If we define P as u = Pu troug a (u,v )=a(u, v ), v V, ten, by te discrete inf sup condition (5), it is trivial to see tat P is a projection operator. In addition, we can bound te norm of te projection operator P as follows. Lemma 2.2. Tere olds P M γ. Proof. See [9] for a simple proof. Now comes te projection error result. Teorem 2.3 (Babuška [7]). Suppose tat bot te continuous problem () and discrete problem (4) are well-posed, ten u u U M inf u w γ w U U. Proof. A standard proof tat uses Lemmas 2. and 2.2 can be found in [0, 9]. Te following best approximation error result immediately follows from Teorem 2.3. Corollary. If M = γ,ten u u U = inf u w w U U. In particular, M = γ = satisfies Corollary. Tat is, if te continuity constant and te discrete inf sup constant are unity, ten te error incurred from te discrete approximation (4) is te best, i.e., it is smallest. Up to tis point, altoug not in te form presented ere, te teory as already been discussed in te DPG metods [, 2, 3]. As can be seen, tere are two spaces to work wit, namely te trial and test spaces, respectively. Te first DPG metod [2] starts wit a given norm in te trial space U, and ten seeks a norm in te test space V so tat M = γ =. In te second DPG metod [3], on te oter and, one defines anorminu from a given norm in V suc tat M = γ =. Clearly, tis is one of te advantages of te DPG metodology since it allows one to coose a norm of interest to work wit wile making te error optimal, i.e., smallest, in tat norm. In tis paper, we explore a different option, namely, we sall not prescribe eiter norms in te spaces U and V. Instead, we let te problem determine its natural energy norms, tus wic norms to be cosen to work wit is out of te question in our new approac. Our single-framework metod will be applied to linear yperbolic equations of advection-reaction type, and we sall sow tat it constructively leads to an existing stabilized numerical metod and a new one. Neverteless, care must be taken since our idea may not be applicable for cases in wic one prefers to work wit particular norms. Te rest of tis section exploits te goal of aving M = γ = to some extent. In particular, we caracterize a few properties for problems aving M = γ =, and ten present sufficient conditions for te infinite dimensional problem to ave M = γ =. Next, we study constructions of finite dimensional subspaces U n and suc tat te well-posedness of te resulting finite dimensional approximation V n

5 WELL-POSED APPROXIMATION METHODS 927 problems is trivially inerited from te infinite dimensional settings. In fact, as sall be sown, our metod automatically delivers unity discrete continuity and inf sup constants, i.e., M = γ =. It sould be pointed out tat te rest of our teory is general so tat it can apply not only to our metod in tis paper, but also te oter DPG metods. We first note tat te inf sup condition (3a) is typically defined by taking first te supremum over te test space V and ten te infimum over te trial space U. However, te following well-known result sows tat as long as te well-posedness, and ence te continuity and inf sup constants, is concerned, te distinction between te test and trial spaces are irrelevant. Tat is, tere is no reason for us to favor one space over te oter. Lemma 2.4. Te problem () is well-posed if and only if γ >0 : inf sup a (u, v) =inf u U v V u U v sup a (u, v) γ. V v V u U v V u U Proof. See [9] for a proof. We first make te following simple observation. Lemma 2.5. Te following are equivalent: (i) M = γ =. a(u,v) (ii) u U we ave u U =sup v V v. V a(u,v) (iii) v V we ave v V =sup u U u. U Proof. By Lemma 2.4, it is sufficient to prove te equivalence of (i) and (ii). (i) (ii): From te continuity condition we ave sup v V a (u, v) v V u U, wic, togeter wit te inf sup condition, implies (ii). (ii) (i): (ii) is equivalent to a (u, v) v V sup v V sup v V a (u, v) v V u U, a (u, v) v V u U, wic implies (i). Te following useful result will be used as guidelines to construct te natural norms in U and V spaces suc tat M = γ =. Teorem 2.6. Suppose te continuity condition olds wit unity continuity constant, i.e., a (u, v) u U v V. Ten tere olds M = γ =if eiter of te following conditions olds: (i) For eac u U \{0}, tereexistsv u V \{0} suc tat a (u, v u )= u U v u V. (ii) For eac v V \{0}, tereexistsu v U \{0} suc tat a (u v,v)= u v U v V.

6 928 TAN BUI-THANH, LESZEK DEMKOWICZ, AND OMAR GHATTAS Proof. We sall sow tat (i) is a sufficient condition for M = γ = to old, and an analogous proof can be done for (ii). It is straigtforward to ave u U = a (u, v u) v u V and Lemma 2.5 concludes te proof. sup v V a (u, v) v V u U, Remark 2.7. In general, te continuity and te inf sup conditions are not related to eac oter, and it is typically more difficult to establis te latter. However, Teorem 2.6 sows tat if te continuity constant is unity and te equality is attainable, ten te continuity condition actually implies te inf sup condition and te inf sup constant is unity as well. Tis important observation is te key result in tis paper and will be exploited trougout for te advection-reaction problem. It sould be pointed out tat if bot conditions in Teorem 2.6 old, ten te 3-tuple (U, V, a (, )) is known as a dual pair. Tat is, te bilinear form a (, ) puts U and V in duality. To te end of te paper, we call te norms in U and V spaces optimal norms if bot continuity and inf sup constants are unity in tese norms. Moreover, we also call te pair u and v u (and ence for u v and v) as te optimal trial and test functions, respectively. Here, optimality is in te sense of Corollary. We are now in position to construct te approximation subspaces U n and V n suc tat te discrete continuity and inf sup constants are unity. Lemma 2.8. Let te assumptions of Teorem 2.6 old respectively for item (i) and (ii) below. (i) Let U n U be a subspace, and construct V n = span { v u V : u U n },a(u,v u )= u U v u V. (ii) Let V n V be a subspace, and construct U n = span { u v U : v V n },a(u v,v )= v V u v U. If te pair of test space V n and trial space U n are constructed by eiter (i) or (ii), ten tere olds M = γ =, and te discrete problem (4) is well-posed if dim U n =dimv n. Proof. Again, it is sufficient to sow item (i). M = is a direct consequence of M =. By construction, we ave, for eac u U n, u U = a (u,v u ) v u V =sup v V a (u,v) v V = sup v V n a (u,v ) v V, wic implies γ =. Te well-posedness of te discrete problem is now clear by te Babuška s teorem. It can be seen tat Teorem 2.6 and Lemma 2.8 do not explicitly specify eiter te optimal test function v u or te optimal trial function u v. A general-purpose approac for coosing an optimal pair of functions is troug te Riesz representation teorem as we now sow. Moreover, if a basis of te trial space U n is specified, we can determine te corresponding basis in te test space and vice versa so tat te finite dimensional problem is well-posed wit M = γ =.

7 WELL-POSED APPROXIMATION METHODS 929 Teorem 2.9. Define te map T : U u Tu V as Tu,v V V = a(u, v). Denote v Tu as te Riesz representation of Tu in V. Suppose a (, ) is continuous wit unity constant and assumption (i) of Teorem 2.6 olds. Take U n U and define V = span {v Tu V : u U n }. Ten, te following old: (i) M = γ =. (ii) Let U n = span {ϕ i} n i=,wereϕ i U, i =,...,n. Ten {v Tϕi } n i= is a basis of V n. Proof. (i) By Lemma 2.8, it is sufficient to sow tat a(u,v Tu )= u U v Tu V. But, by te proof of Teorem 2.6, te definition of norm in V,andte Riesz representation teorem, one readily as, u U =sup v V wic yields, a(u,v) v V =sup v V Tu,v V V v V = Tu V = v Tu V, a(u,v Tu )= Tu,v Tu V V = v Tu 2 V = u U v Tu V. (ii) By virtue of te Riesz representation teorem, v Tu is linear in Tu. Togeter wit te linearity of T, we conclude tat V = span {v Tϕi } n i=.itremains to prove tat te set {v Tϕi } n i= is independent. Assume, on te contrary, tat tere exists i {,...,n} suc tat α i 0and n α i v Tϕi =0 v T( ni= ϕ i) =0, i= wic, by te injectivity of T, implies ( n n α i ϕ i = i= U T α i ϕ i) V =0, i= wic, by te independence of {ϕ i } n i=, in turn implies α i =0, i =,...,n, a contradiction. If we call te result in Teorem 2.9 as te primal approac, ten using Lemma 2.4 we readily ave te following dual analog. Teorem 2.0. Define te adjoint map T : V v T v U as T v, u U U = a(u, v). Assume tat T is injective, i.e., (3b) olds. Denote u T v as te Riesz representation of T v in U. Suppose a (, ) is continuous wit unity constant and assumption (ii) of Teorem 2.6 olds. Take V n V and define U n = span {u T v U : v V n }. Ten, te following old: (i) M = γ =. (ii) Let V n = span {φ i} n i=,wereφ i V,i =,...,n. Ten {u T φ i } n i= is a basis of U n.

8 930 TAN BUI-THANH, LESZEK DEMKOWICZ, AND OMAR GHATTAS In te sequel, we sall not distinguis v u and v Tu as well as u v and u T v since we sall work exclusively wit te Riesz representations. We will use only te primal approac to investigate infinite dimensional settings and ten deduce te properties of te resulting finite dimensional settings. Tat is, we start wit a basis function in te trial space U and ten derive te corresponding optimal basis function in te test space V. Of course, we can also take a dual approac, i.e, we start wit a basis in te test space V andtenseekte corresponding optimal basis in te trial space U. Tis will as well lead to wellposed approximation metods wit M = γ = by Teorem 2.0. However, te approximability is questionable since te resulting finite dimensional trial space is no longer designed to accurately approximate te exact solution. 3. Te first well-posed approximation of linear scalar yperbolic equations In tis section, we discuss in detail ow to apply te primal metod developed in Section 2 to a weak formulation of advection-reaction problems. Te consistency and well-posedness of te infinite dimensional setting are analyzed at lengt. We sall use te Caucy Scwarz inequality to detect te natural norms so tat te bilinear form under consideration is continuous wit M =. Te conditions for equalities to be acievable in te Caucy Scwarz inequality ten allow us to find optimal pair of functions in U and V. In fact, te Riesz representation turns out to be a candidate for equalities to appen. Using tis fact along wit Teorem 2.9 we obtain a finite dimensional approximation metod wit guaranteed well-posedness and M = γ =. 3.. Infinite dimensional setting. Our problem of interest is te first order scalar linear yperbolic equation of te form (6a) (6b) β u + μu = f, in Ω, u = g, on Γ, were Γ = {x Ω :n (x) β < 0} is te inflow boundary; n (x) denotes te outward normal vector at x on te boundary Ω. Assume β [ W, (Ω) ] d wit d {, 2, 3} denoting te dimension of te problem, μ L (Ω), f L 2 (Ω), and g L 2 β n (Γ) wit } L 2 β n (Γ) = {w : w 2L2β n(γ) = β n w 2 dγ <, Γ and u Hβ wit H β (Ω) = { u L 2 (Ω) : β u L 2 (Ω) }. Te following well-posedness of te transport equation (6) is proved in []. Lemma 3.. Assume tat Ω is a Lipscitz domain. Let W = { u H β (Ω) : u Γ =0 } and define W u Tu = β u + μu L 2 (Ω).

9 WELL-POSED APPROXIMATION METHODS 93 Suppose tat β is a filling field, i.e., tere exists a caracteristic line of te vector field β starting from Γ and arriving at (almost everywere) x Ω in finite time. Ten T is a bijective map from W to L 2 (Ω). We partition te domain Ω into N el nonoverlapping elements K j,j =,...,N el suc tat Ω = N el j= K j and Ω = Ω. Here, is defined as = max j {,...,N el }diam (K j ). In addition, we denote by E, wit cardinal number N ed, te set of all unique faces in te mes, namely, te mes skeleton. In tis paper, te term faces is used to indicate eiter edges of 2D elements or faces of 3D elements. Finally, we require β n e L (e) fore =,...,N ed,werenis te normal vector on face e. Multiplying (6a) by a test function v, integrating by parts, and introducing te single-valued flux q L 2 β n (E ) at te element interfaces, we ave, N el (7) [ u (βv)+μuv] dx + Kj \Γβ nqv ds j= K j K j N el = fvdx β ngv ds, j= K j K j Γ wit Kj \Γ denoting te indicator function (also known as te caracteristic function) of K j \ Γ. Clearly, for elements wit caracteristic faces, i.e., β n =0on K j, te boundary integrals corresponding to tese faces simply drop out and q is allowed to be undefined on tese boundaries. Next, integrating by parts one more time gives N el (8) (β u + μu) vdx + β n ( Kj \Γq u ) vds j= K j K j N el = fvdx β ngv ds. j= K j K j Γ If we coose v Kj L 2 (K j ), ten te trace v Kj is not defined. Terefore, we introduce a new ybrid variable r tat lives in te space Π N el j= L2 β n ( K j). Unlike q, wic is single-valued on a face of te skeleton, r is allowed to ave double values depending on te side of tat face. Wit te introduction of r, (8) can be rewritten as N el (9) a (u, v) = (β u + μu) vdx + β n ( Kj \Γq u ) rds K j K j j= N el = l (v) = fvdx β ngr ds, j= K j K j Γ wit u =(u, q), v =(v, r). We define te trial and test spaces as { U = u : u Kj Hβ (K j ) L 2 β n ( K j ),j =,...,N el} = Hβ (Ω ) L 2 β n (E ), V = L 2 (Ω ) Π N el j=l 2 β n (E ).

10 932 TAN BUI-THANH, LESZEK DEMKOWICZ, AND OMAR GHATTAS We first study te consistency of te weak formulation (9). Proposition (Consistency). If u is a solution of (6), ten ( ) u, q = u E is a solution of (9). Conversely, if (u, q) L 2 (Ω) Hβ (Ω ) L 2 β n (E ) is a solution of (9), tenu is a solution of (6). Proof. If u is a solution of (6), ten it is straigtforward to see tat ( ) u, q = u E is a solution of (9). Conversely, let (u, q) U, wit u L 2 (Ω) Hβ n (Ω ), be a solution of (9) for all (v, r) V. First, taking v =0andr L 2 β n (e) weree K j is an internal or an outflow face, for some j {,...,N el}, yields β nq = β nu. Tis implies, for an internal face, β nu = β nu +. Similarly, for an inflow face we obtain β nu = β ng. Next, taking v D(K j )=C0 (K j )andr = 0, ten (9) implies (0) β u + μu = f, in K j, j =,...,N el, in te distributional sense. Since β u + μu, f L 2 (K j ), (0) also olds in L 2 (K j ). Finally, taking v D(Ω), integrating by parts equation (0), and using β nu = β nu + and u L 2 (Ω), we obtain te following identity, uβ vdx = (f μu + βu) vdx, Ω Ω wic implies, in te distributional sense, (βu) =f μu + βu L 2 (Ω), wic in turn implies β u L 2 (Ω). Hence, (0) is in fact valid globally in Ω. Guided by Teorem 2.6, we are seeking norms in spaces U and V suc tat te continuity constant is unity and te equality in te continuity condition is acievable. An approac to obtain tis goal is to apply te Caucy Scwarz inequality, i.e., N el a (u, v) β u + μu L2 (K j ) v L 2 (K j ) j= + Kj \Γq u r L 2 β n ( K j ) L 2 β n ( K j) N el β u + μu 2 L 2 (K j ) + Kj \Γq u 2 2 L 2 β n ( K j) j= }{{} u U N el 2 () v 2 L 2 (K j ) + r 2 L 2 β n ( K. j) j= }{{} v V It is clearly tat te functional V defined above is a norm. Before sowing tat te functional U is indeed a norm, let us ceck weter te equality is attainable.

11 WELL-POSED APPROXIMATION METHODS 933 Since te Caucy Scwarz inequalities are employed, one easily sees tat, given u =(u, q) U, ifv u =(v u,r u ) is defined by (2a) v u = β u + μu, in K j, (2b) r u =sgn(β n) ( Kj \Γq u ), on K j, ten te equality is attainable, i.e., a (u, v u )= u U v u V. Notice tat v u is in fact te Riesz representation of T u (see Teorem 2.9 for te definition of T ). At tis point, we immediately ave te well-posedness of te weak formulation (9). Teorem 3.2 (Well-posedness). Assume tat β is a filling field. Ten, te weak formulation (9) is well-posed wit M = γ =in te norms defined in (). Proof. M = γ = is a direct consequence of Teorem 2.6. Wat remains to be proved is te adjoint injectivity condition (3b). Let v V and assume tat (3) a (u, v) =0, u U. Taking u = 0, ten (3) implies N el β n Kj \Γ qr ds =0, q L 2 β n (E ), j= K j wic in turn implies r =0onE \ Γsincetemap L 2 β n ( K j ) q β nq L 2 β n ( K j ) is surjective due to 0 <c β n L ( K j ) c 2 <. Now, if u W, ten (3) implies (β u + μu) vdx =0, u W, Ω wic, togeter wit te surjectivity of Tu = β u + μu from Lemma 3., yields v = 0 in Ω. Finally, wit v =0inΩandr =0onE \ Γ, (3) becomes N el β n ur ds =0, u Hβ (Ω ). j= K j Γ Since te trace map Hβ (K j) u u Kj L 2 β n ( K j) is surjective, te preceding equation sows tat r =0onΓ. Hencev =0. Having proved te consistency and te well-posedness of te weak formulation (9), we proceed wit finding optimal pairs of trial and test basis functions. For basis functions of te form φ =(0,φ) U, wereφ is a function in L 2 β n (E ), te corresponding basis functions in V for j =,...,N el,aregivenby (4a) v φ =0, in K j, (4b) r φ = Kj \Γφ sgn (β n), on K j. Similarly, for basis functions of te form ϕ = (ϕ, 0), were ϕ Hβ (Ω ), te corresponding basis functions in V for j =,...,N el are given by (5a) v ϕ = β ϕ + μϕ, in K j, (5b) r ϕ = ϕ sgn (β n), on K j. Next, it is natural to substitute te test functions (4) or (5) into (9) to establis equations to solve for te unknowns u =(u, q). Let us proceed wit te generic

12 934 TAN BUI-THANH, LESZEK DEMKOWICZ, AND OMAR GHATTAS test basis in (4) first. If φ is not a zero function on te interface K i K j of elements K i and K j and zero elsewere, ten testing wit v φ defined in (4) te weak formulation (9) becomes β n (q {{u}}) φds=0, φ L 2 β n ( K i K j ), K i K j were {{u}} = u +u + 2 denotes te average. Te positive and negative signs indicate element interior and exterior, respectively. Since (q {{u}}) L 2 β n ( K i K j ), te preceding equation implies q = {{u}} on K i K j. Similarly, taking φ to be a nonzero function on ( Ω \ Γ) K j and zero elsewere, and ten testing (9) wit v φ defined in (4), we obtain, q = u on Ω K j. On te oter and, due to te indicator function Kj \Γ, tere is no test function r φ on te inflow boundary faces. In summary, testing wit te test basis (4), te unknown q on te skeleton is found explicitly as { {{u}} on Ki K (6) q = j u on ( Ω \ Γ) K j. As a result, te trial space can be rewritten as { } U = u : u Kj Hβ (K j ) = Hβ (Ω ), and te norm in U now becomes (7) N el u U = β u + μu 2 L 2 (K j ) + 2 [u] j= 2 L 2 β n ( K j\γ) + u 2 L 2 β n ( K j Γ) wic is essentially te norm in wic te upwind DG metod is stable; see [2], for example. We are now in position to prove tat (7) indeed defines a norm in U. Proposition 2. Assume tat β is a filling field, ten te functional U : U [0, ) defined in (7) is a norm. Proof. Te fact tat te functional U satisfies te positive omogeneity and te triangle inequality is obvious. Terefore, it remains to prove tat u U = 0 implies u = 0. We first consider elements wose faces are subsets of te inflow boundary. In tis case, u U = 0 implies (8a) β u + μu =0, in K j, (8b) u =0, on K j Γ, (8c) [[u] = 0, on K j \ Γ. By Lemma 3., u =0inK j is te unique solution of (8a) and (8b). Since β is a filling field, K j \Γ must be inflow boundaries of oter adjacent elements. However, (8c) implies tat u = 0 on tese inflow boundaries. By induction, we conclude tat u = 0 is zero on Ω, and tis ends te proof. 2,

13 WELL-POSED APPROXIMATION METHODS 935 (9) Next, substituting te optimal test basis function (5) into (9) we ave N el (β u + μu)(β ϕ + μϕ) dx β n ( Kj \Γq u ) ϕds K j K j j= N el = f (β ϕ + μϕ) dx + β n gϕ ds. j= K j K j Γ We can simplify (9) furter by using te explicit value of q in (6), renaming ϕ to v, and rewriting te bilinear and linear forms in (9) as N el a (u, v) = (β u + μu)(β v + μv) dx j= K j + β n [u] vds+ β n uv ds 2 K j \ Ω K j Γ N el (20) l (v) = f (β v + μv) dx + β n gv ds, j= K j K j Γ were te jump operator [u] = u u + is employed. It is important to point out tat te weak formulation (20) is completely equivalent to (9) except tat te auxiliary ybrid variables q and r are now eliminated, and tat te trial and test spaces are now te same, i.e., U = V = Hβ (Ω ) Finite dimensional setting and convergence analysis. Now, we select te same finite dimensional subspace for bot trial and test functions, i.e., V n = U n = span {ϕ i U, i =,...,n}. Te consistency of te weak formulation (9) on finite dimensional subspaces U n and V n is automatically inerited from Proposition. On te oter and, using Lemma 2.8 we immediately ave te well-posedness of te finite dimensional approximate problem. Teorem 3.3. Let te bilinear form a (, ) and te linear form l ( ) be defined as in (20). Te following problem, { Seek u U n suc tat a(u,v )=l(v ), v V n, is well-posed wit M = γ =. Tat is, our effort in Section 2 is now paid off by te trivial well-posedness of te finite dimensional problem, wic is typically not using oter metods [6]. In order to use polynomial approximation results [3, 4, 5, 6], we specify te finite dimensional subspaces U n and V n { as U n = u Hβ (Ω ):u Kj P p j (K j ),j =,...,N el} = V n, were P p denotes te polynomial spaces of order at most p. For d {2, 3}, P p could be te usual polynomial spaces for triangular and tetraedral meses, and te tensor product polynomial spaces for quadrilateral and exaedral meses.

14 936 TAN BUI-THANH, LESZEK DEMKOWICZ, AND OMAR GHATTAS At tis point, a closer look tells us tat our simplified weak formulation (20) is one of te existing least squares DG (LSDG) metods [7]. However, it is important to empasize tat, for te DG metods in general, and te oter LSDG metods in particular, one usually starts by introducing a numerical flux (typically borrowed from finite volume metods). One ten defines bilinear and linear forms (adoc typically) over finite dimensional polynomial subspaces, and proves te well-posedness of te resulting finite dimensional approximate problem wit some prescribed norm. Here, starting from te requirement of aving unity continuity and inf sup constants, and coosing te weak formulation (9) to apply our teory developed in Section 2, we constructively (and accidentally) derive an LSDG metod from a well-posed infinite dimensional setting. Te distinct feature of our metod is tat te flux is introduced as a new unknown, and ten found by fulfilling stability. It sould be pointed out tat our teory is general in te following sense. If it is applied to different weak formulations, one will constructively obtain different numerical metods, again, wit trivial well-posedness for te finite dimensional approximation problem, as we sall sow in te next sections. Since our resulting approximation metod coincides to an LSDG approac, te following p-convergence result is immediate from [7]. Teorem 3.4. Suppose tat u Kj H s j (K j ),s j > 2,j =,...,Nel and te mes is affine. Ten, tere exists a positive constant C, independent of j = diam (K j ), p j,andu suc tat el N u u 2 U C 2σ j 2 j j= p 2s u 2 j 2 H s j (K j ), j wit p j and 0 σ j min (p j +,s j ). Furtermore, if te mes is quadrilateral or exaedral and te exact solution u is elementwise analytic, ten te following estimation olds: el N u u 2 U C ( ) 2σj 2 j p j e 2b jp j meas (K j ), 2 j= were C depends on u, d, and te sape-regularity, b j depends on d, andmeas (K j ) denotes lengt, area, and volume of K j for d =, d =2,andd =3, respectively. Proof. A proof can be found in [7] and te references terein. Remark 3.5. By Corollary, te error estimation is optimal in bot and p. In particular, if te exact solution is elementwise analytic, our approximation metod delivers exponential convergence in p. 4. Te second well-posed approximation of linear scalar yperbolic equations In te first application of our abstract teory for linear scalar yperbolic equations in Section 3, we ave cosen te weak formulation (9) obtained after integrating (6) by parts twice. In tis section, we apply our teory to te weak formulation (7) obtained after integrating (6) by parts once. As will be sown, we obtain a different well-posed infinite dimensional setting, ence leading to different finite dimensional approximation wit guaranteed well-posedness and M = γ =.

15 WELL-POSED APPROXIMATION METHODS Infinite dimensional setting. Our starting point is te weak formulation (7) obtained by integrating (6) by parts once. Tis formulation can be written in a more compact form as (2) N el u [ (βv)+μv] dx + β nq [[v ] ds K j 2 K j j= N el = fvdx K j j= K j Γ β ngv ds, were on te outflow boundary we conventionally define v + = v andonte inflow v + = v. We define te trial and test spaces as { } U = u : u Kj L 2 (K j ) L 2 β n ( K j ) = L 2 (Ω ) L 2 β n (E ), { } V = v : v Kj Hβ (K j ) = Hβ (Ω ). Again, guided by Teorem 2.6, we are looking for natural norms in spaces U and V suc tat te continuity constant is unity and te equality in te continuity condition is acievable. Similar to Section 3, we apply te Caucy Scwarz inequality to obtain, N el a (u,v) (βv)+μv L2 (K j ) u L 2 (K j ) j= (22) + 2 q L 2 β n ( K j) [[v ] L 2 β n ( K j ) N el 2 (βv)+μv 2 2 L 2 (K j ) + 2 [v ] j= L 2 β n ( K j) }{{} v V N el 2 u 2 L 2 (K j ) q. j= L 2 β n ( K j) }{{} u U Equalities in (22) are acievable if we use te Riesz representations, i.e., given u =(u, q) U, (23a) (23b) u = (βv u )+μv u, in K j, q =sgn(β n)[[v u ], on K j. Once we know tat te equality is obtainable under condition (23), te consistency and well-posedness of (2) wit respect to te norms defined in (22) similarly follows Proposition and Teorem 3.2. We next use (23) to find optimal pairs of trial and (corresponding) test basis functions. For basis functions of te form φ =(0,φ) U, wereφ is a function in

16 938 TAN BUI-THANH, LESZEK DEMKOWICZ, AND OMAR GHATTAS L 2 β n (E ), te corresponding basis functions in V are given by, for j =,...,N el, (24a) (24b) (βv φ )+μv φ =0, in K j, [[v φ ] = sgn (β n) φ, on K j. Similarly, for basis functions of te form ϕ =(ϕ, 0) U, wereϕ L 2 (Ω ), te corresponding basis functions in V are given by, for j =,...,N el, (25a) (βv ϕ )+μv ϕ = ϕ, in K j, (25b) [[v ϕ ] = 0, on K j. Once te test functions are found using (24) or (25), we can substitute tem into (2) to establis equations to solve for te unknowns u =(u, q). Let us proceed wit te generic test basis in (24) first. If φ is different from zero on e E and zero elsewere on te skeleton, ten testing (2) wit v =(0,v φ ) yields, (26) β n qφds= fv φ dx β n gv φ ds. e Ω Γ As can be seen in (26), for eac φ L 2 β n (e),e E, q L 2 β n (e) can be locally solved independently of u. Now if ϕ Kj is a nonzero function in K j but zero elsewere, ten testing (2) wit v =(0,v ϕ ) gives, (27) uϕ dx = fv ϕ dx β n gv ϕ ds, K j Ω Γ wic sows tat te unknown u can also be computed locally element-by-element and independently of q Finite dimensional setting and convergence analysis. Now, given a finite dimensional subspace U n = span {u i,i=,...,n} U, were { φi =(0,φ u i = i ) i =,...,m, ϕ i =(ϕ i, 0) i = m +,...,n, were φ i and ϕ i are local functions previously used to obtain (26) and (27). Te corresponding optimal finite dimensional test space is given by V n = span {v ui,i=,...,n}, wit basis vectors v ui computed as in (24) or (25). Te consistency of te weak formulation (26) and (27) on finite dimensional subspaces U n and V n is automatically inerited from te infinite dimensional setting. On te oter and, using Lemma 2.8 we immediately ave te well-posedness of te finite dimensional approximate problem similar to Teorem 3.3. Teorem 4.. Let te bilinear form a (, ) and te linear form l ( ) be defined as in (26) and (27). Te following problem, { Seek u n =(u n,qn ) U n suc tat a(u n, vn )=l(vn ), vn V n, is well-posed wit M = γ =.

17 WELL-POSED APPROXIMATION METHODS 939 It is important to point out tat te computation of eac test basis function requires a solution of te adjoint yperbolic equation (24) or (25). Since β, and ence β, is assumed to be a filling field, it follows tat starting from an element K j, it is always possible to identify te neigbor elements, called outflow elements of K j, wose inflow faces (wit respect to te adjoint flow) are te outflow faces of K j. We continue tis procedure wit te current outflow elements, and by induction, we can marc to te inflow boundary in finite time since tere is a finite number of elements in te mes. In oter words, te adjoint flow starting from an element K j must arrive at te inflow boundary Γ by mass conservation in finite time for any triangulation. Once tese elements are found, tey form a streamtube starting from K j to te inflow boundary. Note tat a streamtube is allowed to ave elements wose faces are bot inflow and outflow wit respect to te adjoint velocity field β. Terefore, we can eiter marc te solution for an optimal test function element by element along a streamtube or solve for it on all elements simultaneously. Once te finite dimensional space V n is constructed, te weak formulation (2) becomes decoupled weak formulations (26) and (27). Tis allows us to locally compute te trace q face-by-face, and te approximate solution u element-by-element, independently. Te solution procedure can be divided into offline and online stages as follows. Since te computation of te test basis functions does not depend on te boundary condition g and te forcing f, it is done once for te offline stage. Furtermore, solving for eac optimal test function is an independent task, and ence can be done in parallel. As a result, te offline cost is in fact te same as te cost of solving for one optimal test function if te parallelization is fully exploited. After tat, te test basis functions can be used for any g and f, wile preserving te best approximation property. Terefore, in te online stage, our resulting finite dimensional approximation metod provides fast and best (in te U -norm) solution u for any admissible g and/or f. It is fast because we only need to locally solve for u element-by-element by inverting te elemental mass matrices as sown in(27). Itisbestinte U -norm due to Corollary. We sould point out tat te forcing f is typically projected onto te subspace span {ϕ i,i= m +,...,n}, and te boundary condition g onto span {φ i Γ,i=,...,m}. Tus, te rigt side of (27) can be evaluated once in te offline stage for te basis functions in subspaces span {ϕ i,i= m +,...,n} and span {φ i Γ,i=,...,m}. Clearly, if f =0, te online stage is extremely fast and negligible in cost. Tis is particularly useful for real-time source or boundary condition inverse problems using optimization metod in wic one as to solve te forward equation (6) many times wit different g and/or f. Moreover, our numerical results sow tat te offline cost may be muc more tan offset by te optimality and accuracy wit no nonpysical diffusion of our metod over te popular upwind DG approaces. Te above offline online procedure can also be seen as a new model reduction (also known as reduced-order modeling) approac. Typical projection-based model reduction approaces [8, 9] begin wit a finite, but large, dimensional approximation problem of dimension n, e.g., (4). Wen n is very large (i.e. for large-scale problems), te discrete problem (4) is intractable for real-time simulation, inverse problems, optimal control, uncertainty quantification, etc. [8, 20]. Te idea beind model reduction is to seek a subspace U n r = {ξ i,i=,...,n r } U n, n r n,

18 940 TAN BUI-THANH, LESZEK DEMKOWICZ, AND OMAR GHATTAS were te basis functions ξ i,i =,...,n r typically ave global support. Te discrete problem (4) is now solved for te pair {U n r,vn r } instead of {U n,vn },wic is muc ceaper since it as only n r n unknowns. Clearly, one as to address te well-posedness of te reduced problem and its accuracy. Compared to te existing model reduction tecniques, our reduction tecnique is more expensive in te offline stage. In te online stage, te cost of constructing an approximate solution is more or less similar, namely, O (n). However, te distinct feature of our metod is tat it is a direct model reduction tecnique. Tat is, it does not require to construct te reduced spaces U n r and V n r. Moreover, te well-posedness of our direct model reduction metod is trivial wit M = γ = by Lemma 2.8, and our reduced solution u n is te best in te space U n due to Corollary. In addition, we always ave guaranteed optimal p-convergence result for te reduced solution u n,aswe sall sow momentarily. Anoter important feature of our approximation is wort pointing out. Tat is, te test basis vectors computed from (24) or (25) are continuous across elements wile te trial basis is completely discontinuous, implying discontinuous solution u in general. Tus, our approximation metod can be considered as a new discontinuous finite element metod wic is in between continuous finite element metods [2] (in wic bot trial and test spaces are continuous), and te DG metods [22] or te DPG metods [, 2, 3] (in wic bot trial and test spaces are discontinuous). It also as te flavor of te ybridized discontinuous Galerkin metod [23] due to te present of te ybrid variable q. For ease in writing, we will also call our metod a DPG metod. We next coose finite dimensional polynomial spaces and discuss te convergence of te resulting discontinuous finite element metod. To begin, we specify { U = u =(u, q) U u Kj P p j (K j ), j =,...,N el }, q e P p e (e), e =,...,N ed. We first recall te following useful result on polynomial approximation teory [3, 4, 5, 6]. For u H s (D), were D could be an element in te mes (i.e., D = K j ) or its faces (i.e., D K j ), tere exists Πu P p (D) suc tat (28) u Πu 2 L 2 (D) C 2σ p 2s u 2 H s (D), s 0, were σ =min{p +,s}, and =diam(d). Ten, we ave te following convergence result wose proof is almost trivial. Teorem 4.2. Suppose tat u Kj H s j (K j ),s j > 2,j =,...,Nel,andq e H s e (e),e =,...,N ed. Let te mes be affine. Ten, tere exists a positive constant C, independent of j =diam(k j ), e =diam(e), p j, q, andu suc tat N el u u n 2 U C 2σ j j p 2s u 2 j H s j (K j ) + e p 2s q 2 e j e H s e (e), j= e K j,e E 2σe wit p j,p e, 0 σ j min (p j +,s j ),and0 σ e min (p e +,s e ). Proof. Te proof is obvious using te definition of te norm in (22), te best approximation error in Corollary, and te approximation error (28).

19 WELL-POSED APPROXIMATION METHODS Numerical results In tis section, we present several numerical results to support our findings. Since our first metod coincides wit a least squares DG described by Houston et al. [7], we refer te readers to tis paper for extensive numerical results. We will terefore sow only numerical results for our DPG metod described in Section 4. Furtermore, since we are only interested in u n, wic is independent of qn, we will ignore te computation of q n. 5.. One-dimensional example. We consider a one-dimensional example in wic te optimal test functions can be computed analytically. In particular, we coose Ω = (0, ), β =,μ =0,g =arctan( 00), and te forcing function f to be 00 f(x) = (x ) 2. Under tese assumptions, te exact solution can be found as u (x) = arctan [00 (x )]. For one-dimensional problems, our mes is simply given by 0 = x 0 <x <... < x N el =,andk j =(x j,x j ),j =,...,N el. Let us denote F Kj (r), r I= [, ] as a diffeomorpic map from te master element I to K j. From (25), te optimal test function is found by integrating from te outflow to te inflow boundaries, i.e., v ϕ = xj x ϕdx, j =,...,N el, [v ϕ ] = 0, at x j and x j. If supp ϕ K j,andϕ F (r) =P m (r) istemt order Legendre polynomial, ten we ave, r 2 dp m (29) v Pm (r) = J Kj m (m +) dr, m 0, were J Kj is te Jacobian of te map F Kj. From (25) and (29) we conclude tat v Pm (r) as local support since v Pm ( ) = v Pm () = 0, in particular, supp v Pm (r) K j, m 0. Form =0,weobtain 0 in K i, i > j, v P0 (r) = J Kj ( r) in K j, 2J Kj in K i, i < j. We ave sowed tat most of te optimal test functions ave locally compact supports. Tis makes te evaluation of te rigt side of (27) negligible. Moreover, te mass matrix on te left side of (27) becomes diagonal due to te ortogonality of te Legendre polynomials. Tis makes our metod extremely efficient in te online stage since tere is no need to invert any matrices to solve for u n, no matter ow fine te mes is. For te numerical results, we coose uniform order p for all elements. We will compare our DPG metod wit te original upwind DG [24, 25]. In Figure, we plot te DPG and DG solutions for various polynomial orders on a uniform mes wit four elements. In te region wit small solution gradient, bot metods are comparable. However, in te steep gradient region, te DPG solutions are less

20 942 TAN BUI-THANH, LESZEK DEMKOWICZ, AND OMAR GHATTAS u,u n DPG DG exact x u,u n DPG DG exact x (a) p = (b) p = 2 u,u n DPG DG exact x u,u n DPG DG exact x (c) p = 3 (d) p = 4 Figure. Te DPG and DG solutions for various orders p =, 2, 3, 4 wit =0.25. Te mes as four elements. oscillatory tan te DG ones. As can also be seen, te DPG solutions ave smaller undersootings and oversootings. Tis is due to te fact tat te DPG solutions minimize te error in te energy norm (22), a component of wic is te error in te L 2 norm. Terefore, oscillations wit big amplitude tat ave significant L 2 norm are not allowed in te DPG solutions. Te DG metod, owever, does not ave suc a property. Figure 2 plots te L 2 error for refinements in te log-log scale. We approximate te convergence rate by least squares fitting te convergence curves wit straigt lines. As can be observed, not only are te DPG solutions more accurate by an order of magnitude, but also tey ave better convergence rates. Figure 3 sows te L 2 error for p refinements in te linear-log scale. Bot DPG and DG exibits exponential convergence rate, wic is consistent wit Teorem 3.4. Again, for te same polynomial order and number of unknowns, te DPG metod is more accurate tan te DG metod.

21 WELL-POSED APPROXIMATION METHODS p = p =2 0 4 p = p =4 p =5 error in L 2 norm (a) DPG metod 0 2 p = p =2 p =3 p =4 error in L 2 norm p = (b) DP metod Figure 2. -convergence rate: 2(a) Log-log scale plot of te error of our DPG metod in te L 2 -norm, i.e., u u n L 2 (Ω ) ; 2(b) Loglog scale plot of te error of te nodal DG metod in te L 2 -norm. Te mes is refined in for different polynomial orders from p = to p = 5. Te convergence is sown for four different mes sizes, = {0.43, 0.22, 0., 0.054}.

22 944 TAN BUI-THANH, LESZEK DEMKOWICZ, AND OMAR GHATTAS =0.06 error in L 2 norm = = = p (a) DPG metod =0.06 error in L 2 norm = = = p (b) DG metod Figure 3. p convergence: 3(a) Linear-log scale plot of te error of our DPG metod in te L 2 -norm, i.e., u u n L 2 (Ω ) ; 3(b) Linear-log scale plot of te error of te nodal DG metod in te L 2 -norm. Temesisrefinedinp, and te convergence is sown for four different mes sizes, {0.06, , , 0.002}.

23 WELL-POSED APPROXIMATION METHODS Two-dimensional examples. For te first example, we coose Ω = (, ) 2, β =(0.8, 0.6), μ = 0, and te forcing f = π ) (π 0 ( + y)[+y +.5(+x)] cos ( + x)(+y) 2 /8, so tat te exact solution is u =+sin ( ) π ( + x)(+y) 2 /8. Te boundary condition is simply te restriction of te exact solution on te inflow boundaries { y =0, x, g = x =0, y. For tis problem, even toug it may be still possible to compute te optimal test functions exactly, a more general and practical approac is to solve for tem approximately. In particular, we coose to approximate te optimal test functions using te following finite dimensional test subspace { V Δp = v V : v Kj P p j+δp j,j =,...,N el} V. Clearly, V Δp asymptotically approaces V as Δp j, owing to te density of te space of polynomials. We now solve (25) for v ϕ by marcing along te streamtube starting from K j to te inflow boundary Γ using te original DG metod [24, 25], wic seems to be natural for tis problem. Figure 4 sows two streamtubes starting from two different elements (black) to te inflow boundary Γ. Clearly, te support of a streamtube, and ence te corresponding optimal test functions, can be significant if te mes does not align wit te streamlines. Alternatively, instead of carrying out te computation on te wole streamtube, one can make furter approximation by creating a line of elements starting from K j to Γ, and tis issue will be addressed in our future work [26]. Figure 4. Two streamtubes starting from two different elements (black) to te inflow boundary Γ. Once te optimal test functions v ϕ are computed, te computation of te approximate solution u on eac element via (27) is ten followed by inverting te local mass matrix. Figure 5 presents te -convergence of te DPG solution for various values of te enriced exponent Δp j = {0,, 2, 3} wit p j =4,j =,...,N el. As can be observed, Δp j = 0 is te least accurate coice, and te accuracy does not seem to increase for Δp j greater tan unity. For tis reason, Δp j = will be used from ere to te rest of tis section.

24 946 TAN BUI-THANH, LESZEK DEMKOWICZ, AND OMAR GHATTAS error in L 2 norm Δ p = 0 Δ p = Δ p = 2 Δ p = Figure 5. Te DPG solutions for various values of te enriced exponent Δp j = {0,, 2, 3} wit p j =4,j =,...,N el. Figure 6 sows tat te convergence rate of te DPG metod is optimal in and exponential in p. We next consider a more callenging problem in [2] on Peterson s meses [27]. For tis problem, we specify Ω = (0, ) 2, β =(0, ), μ =0,f = 0, and te inflow boundary condition as g = u (x, 0) = sin (6x), x (0, ), so tat te exact solution is u (x, y) = sin(6x). In Figure 7 we compare te -convergence rate of te DPG metod wit tat of te upwind DG metod [24, 25]. Te DPG metod delivers optimal convergence rates for all polynomial orders wile te DG metod as sarp sub-optimality. Te sub-optimality of te DG metod is well known [27], and te numerical result in Figure 7(b) interestingly indicates tat it seems to appen only for polynomial orders below tree. As can also be observed, te DPG is more accurate tan te DG for all -andp-cases, and tis is a direct consequence of te best approximation property in Corollary.

25 WELL-POSED APPROXIMATION METHODS p = p =2 p =3 error in L 2 norm p = (a) DPG -convergence =0.5 error in L 2 norm =0.25 =0.2 = p (b) DPG p-convergence Figure 6. DPG - and p-convergence rates: Log-log scale plot of te error of te DPG metod in te L 2 -norm, i.e., u u n L 2 (Ω ); Te mes is refined bot in and p for = {0.5, 0.25, 0.25, } and for p = {, 2, 3, 4}.

26 948 TAN BUI-THANH, LESZEK DEMKOWICZ, AND OMAR GHATTAS p =0 p = error in L 2 norm p =2 p =3 p = (a) DPG -convergence p =0 p = p =2 error in L 2 norm p =3 p = (b) DG convergence Figure 7. -convergence rate: 7(a) Log-log scale plot of te error of te DPG metod in te L 2 -norm, i.e., u u n L 2 (Ω ) ; 7(b) Loglog scale plot of te error of te nodal DG metod in te L 2 -norm. Te mes is refined in for different polynomial orders from p =0 to p = 4. Te convergence is sown for four different mes sizes = {0.663, , 0.046, }.

27 WELL-POSED APPROXIMATION METHODS 949 Since te DPG metod can be considered as a least-squares metod in te dual space V [2], we would like to study its diffusion property numerically. To tis end, te next example is extracted from [7], in particular, Ω = (0, 2) (0, ), β =(+sin(πy/2), 2), μ =0,f = 0, and te inflow boundary condition x =0, 0 y, g = sin 6 (πx) 0 <x,y =0, 0 x 2,y =0, for wic te exact solution can be found using te metod of caracteristics. Figure 8 compares te solutions using te least-squares DG metod in Section 3, te upwind DG, and te DPG in Section 4. Te computational mes is sown in Figure 8(a), and te solution order p = is cosen uniformly for all elements K j, j =,...,N el. Te results sow tat te least-squares DG is overly diffusive, te DG is less diffusive, and te DPG as te least diffusion. In addition, te DPG metod is te most accurate one. It terefore indicates tat wile standard leastsquares metods in primal spaces are very diffusive, tose in dual spaces evidently do not seem to be te case. To furter confirm tis, we consider te following simple example [2] in wic we specify Ω = (0, ) 2, β =(0, ), μ =0,f =0,and te inflow boundary condition as g = u (x, 0) = x 2, x (0, ). Te p = 0 solutions using te upwind DG and te DPG metods are sown in Figure 9. Clearly, te DPG solution does not seem to ave any noticeable cross-wind diffusion (i.e., te colors do not diffuse orizontally along te y-direction), wereas te upwind DG solution does. Tis reflects te fact tat wile most of numerical metods for yperbolic equations introduce numerical diffusion eiter explicitly or implicitly (e.g., troug te numerical fluxes as in many DG metods) to gain stability, and ence introduce nonpysical diffusion, te DPG stability comes directly from te functional setting troug te Banac-Nečas-Babuška teorem on te infinite dimensional level. We ave solved for te optimal test functions in te subspace V Δp V rater tan V. Terefore, te inerited well-posedness wit te optimal error estimate of te finite dimensional approximation problem does not generally old, even toug our numerical results sow tat it is in fact te case. Fortunately, a recent work [28] sows tat te ineritance is still guaranteed under some suitable conditions for Laplace and linear elasticity equations. A similar result for our DPG metod is a subject of our future work. We empasize again tat our DPG metod does not require any equation solves on te online stage if ortogonal polynomials or mass lumpings (e.g. collocation on Legendre-Gauss-Lobatto nodes) are used, and ence (partially) compensating for te cost of computing te optimal test functions. Work is in progress to minimize te test function supports in two- and tree-dimensional problems [26]. We sould mention tat streamline meses [29] may be an effective option in reducing te cost and complexity of te offline stage, since one only needs to solve for te optimal test basis functions along predefined streamtubes, accordingly. Streamline meses also imply tat te support of any optimal test basis function is confined inside a streamtube, allowing us to evaluate te rigt side of (27) only along streamtubes, and ence making te online stage even faster.

28 950 TAN BUI-THANH, LESZEK DEMKOWICZ, AND OMAR GHATTAS Least squares DG exact (a) Computational mes (b) Least-squares DG solution upwind DG exact DPG exact (c) Upwind DG solution (d) DPG solution Figure 8. Solution at te outflow wit p j =,j =,...,N el on 0 0 2,y = : 8(a) Te mes; 8(b) Te least-squares DPG metod in Section 3; 8(c) Upwind DG metod; 8(d) Te DPG metod. Solid lines are te exact solution, and numerical solutions are dased lines. Figure 9. Numerical solutions wit p j =0,j =,...,N el : 9(a) p = 0 DG solution; 9(b) p = 0 DPG solution.