Abstract
We present an extension of the discontinuous Galerkin method which is applicablc to the numerical solution of diffusion problems. The mcthod involves a weak imposition of coni inuity condit ions on the solution values and onlluxes across interelement boundaries. \Vithin each element, arbitrary spectral approximations can be constructed with different orders p in each element. We demonstrate that the method is elemenlwise conservative. a property uncharacteristic of high-order finite clements.
Abstract
for clarity. we focus on a model class of linear second-order boundary value problems. and we develop a priori error estimates. convergence proofs. and stability estimates. The results of numerical experiments on It-and I'-convergence rates for representative two-dimensional problems suggest that the method is robust and eapable of delivering exponential rates of convergence.
Abstract
:c I'NX ,v.
Abstract
• d:·no;" p",,,
Abstract
Key llords: discontinuous galcrkin: tinitc elcments.
Abstract
This paper presents a new type of discontinuous Galerkin method (DGM) that is applicable to a broad class of partial di frerential equations. In particular. this paper addresses the treatment of diffusion operators by tinite element techniqucs in which both the approximate solution and the approximate fluxes can experience discontinuitics across inrerelcmcnt boundaries. Among features of the method and aspects of the study presented here are the following:
Abstract
• 1I priori error estimates are derived so that the parameters affecting the rate of convergence and limitations of the method are established;
Abstract
• the method is suited for adaptive control of error ami can deliver high-order accuracy where the solution is smooth:
Abstract
• the method is robust and exhibits clementwise conservative approximations: Copyright I:.£) lINg by Academii.' Prl'ss All rights of rc.prodUl.:tioll in any foml n:sl:rvl.."d.

• adaptive versions of the method allow for near optimal meshes to be generated;

• the cost of solution and implementation is acceptable.

The use of dis<:ontinuous tinite element methods for se<:ond-and fourth-order elliptic problems dates back to the early 1960s. when hybrid methods were developed by Pian an his collaborators.

The mathematical analysis of hybrid methods was done by Babuska [7] .

Bahuska et al. [t) . 101. and others. In 1971, Nitsche [38] introduced the concept of replacing the houndary multipliers with the normal fluxes and added stabilization lerms to produce optimal convergenL:e rates. Similar approaches can be traced back to Ihe work of Percell and

Wheeler 140[, Wheeler [421, and Arnold [3] . A different approach was Ihe /I-formulation [3] utilize the hilinear 1'01111 of the GEM augmented with a penalty term which includes the jumps of the solution across clements. The disadvantages of the last approach indude the dependence of stabilily and convergence rates on the penalty parameter. the loss of the conservation property at clement level. and a bad conditioning of the matrices. The DGM for diffusion operators developed in Ihis study is a modification of the GEM. which is free of the above deficiencies. More details on these formulations and Ihe relative merits of each one will he presented in Section 3.1.

The lirst study of discontinuous finite element methods for linear hyperbolic problems in two dimensions was presented by Lesaint and Ravial1 in 1974 [311 (see also [30.32] ). who derived a priori error estimates for special cases. Johnson and PitkUranta 1281 and Johnson [27] presented optimal error estimates using mesh-dependent n0I111S. Among the applications of the discontinuous Galerkin method to nonlinear first-order systems of equations, Cockbul11 and Shu [19] [20] [21] [22] [33] [34] [35] [36] and Warburton et at. [41] solved the NavierStokes equations discretizing the Euler Iluxes with the DGM and using a mixed fonllulation for the viscous fluxes. A similar approach was followed by Cockburn and Shu in the developmenl of the local discontinuous Galerkin method [231; see also short-course notes by Cockbull1 r 18].

In the present investigation. a discontinuous Galerkin method for second-order systems of partial differential equations is presented in which the solution and its derivatives are discontinuous across element bnundaries. The resulting scheme is elementwise conservative, a property not common to tinite element methods. pm1icularly high-order methods. The formulation supports 11-. p-, olld lip-version approximations and can produce sequences of approximate solutions that are exponentially l:onvergent in standard nonns. We explore the stabi Iity of thc method for one-and two-dimensional model problems and we present a priori error estimates. Optimal order 11-and fJ-convergenn~in the lit norm is observed in one-and two-dimensional applications. Following this Introduction, various mathematical preliminmies and notations are presented in Section 2. Section 3 presents a variational formulation of a general linear di ITusion problem in a functional selling that admits disl:ontinuities in ftuxes and values of the solution across subdomains. Here properties of this weak fonllulation are presented, laying the groundwork for discontinuous Galerkin approximations that are taken up in Section 4. In Section 4, the discolltinuous Galerkin method for diffusion problems is presented. A study of the stability of the method is presented and a priori error estimates are derived. These theoretical results are then conlirmed with llullIerical experiments. The results of several applications of the method to two-dilllcnsionallllodel problems are recorded and discussed. It is shown. among other features. that exponential rates of cnnvergence can be attained. where S E L2(Q). and A E (L OC(Q))"'" is a dilTusivity matrix characterized as

O'taT888a:dbb7:2d49:b3f6:8605:bf4e:caed:e5a4888a:dbb7:2d49:b3f6:8605:bf4e:caed:e5a4 afA(x)888a:dbb7:2d49:b3f6:8605:bf4e:caed:e5a4888a:dbb7:2d49:b3f6:8605:bf4e:caed:e5a4

O'oa T a. at888a:dbb7:2d49:b3f6:8605:bf4e:caed:e5a4888a:dbb7:2d49:b3f6:8605:bf4e:caed:e5a4 0'0> O. Va E ffi.".

a.e. x in Q.

The boundary aQ consists of two disjoint paris. rn on which Dirichlet conditions are imposed. and rt\ on which Neumann conditions arc imposed: I'D n r.'1= Vl.fD U rN = aQ, and meas fD > O. whereas boundary conditions are

Unfortunately. the classical statement ( 1 )- (2) 
2,3. Standard Weak Formulation and (;alerkin Approximation
A classical weak formulation of problem (I )- (3) is stated as follows:
2,3. Standard Weak Formulation and (;alerkin Approximation
./0.
2,3. Standard Weak Formulation and (;alerkin Approximation
./Q./t:< (4) and ii E HI (Q) is such lhat yoii II'll = .t. Yo being the trace operator. A weak solution of
2,3. Standard Weak Formulation and (;alerkin Approximation
The existence of so]utions to (4) can be established using the classical generalized LaxMilgram theorem IS. 391.
2,3. Standard Weak Formulation and (;alerkin Approximation
A Galerkin approximation of (4) consists of constructing families of closed (generally finite-dimensional) subspaces. \I" C V, and seeking solutions It" E V" to the following problems:
2,3. Standard Weak Formulation and (;alerkin Approximation
Find u" E VII such that l'eh.
2,3. Standard Weak Formulation and (;alerkin Approximation
and
2,3. Standard Weak Formulation and (;alerkin Approximation
A straightforward t:alculation reveals that the enor U -Uh in the approximation (5) of (4) satisfies the estimate
2,3. Standard Weak Formulation and (;alerkin Approximation
where M is the continuity constant defined aB
2,3. Standard Weak Formulation and (;alerkin Approximation
Proofs of the generalized Lax-Milgram Iheorem and of the estimate (X) can be found in [5, 39] .
Families of Regular Partitiuns
Since our discontinuous approximations are to be ultimately defined on partitions of the domain a, we now establish notations and conventions for families of regular partitions
Families of Regular Partitiuns
[17]. Let P = (Ph (Q) lh>o be a family of regular partitions of a c IR
Families of Regular Partitiuns
subdomains ar (see Fig. 1 operator [·1 defined on r,'f = QI' n Q I '# til as ( 12) and the average operator (-) for the nonllal flux is defined for (A V v) . n E L2( r,r) as (13) where A is the dilTusivity. Note that n represents the outward normal from the elcmcnt with higher index.
Polynomial Approximations on Partitions
For future reference. we record a local approximal ion property of polynomial finite elemeIll approximations. Let Q be a regular master element in IR". and let IFn ..J be a family of invertiblc maps from Q onto 0.,. (see Fig. 2 \ip(P,,) = IIp/>.(0.,.).
Polynomial Approximations on Partitions
N (Pit) being the number of elements in partit ion Pt,.
Polynomial Approximations on Partitions
The approximation properties of "1'(Ph) will be estimated using standard local approximation cstimatcs (see 16]). Let /I E H·'(0.,.): therc exist a constant C depending on s and on the angle condition of 0.", but independent of II. he = diam(0.,.). and P", and a polynomial "I' of degree P", such that for any 888a:dbb7:2d49:b3f6:8605:bf4e:caed:e5a4: /' 888a:dbb7:2d49:b3f6:8605:bf4e:caed:e5a4: s the following estimate holds:
Polynomial Approximations on Partitions
here II· lid'!. denotes the usual Sobo\ev nOllll. and IL = min(p" + I . .1').
Polynomial Approximations on Partitions
The following local inverse ine4ualities for a generic element 0.e arc valid for U E P P., (0.e) and Pe > 0 (sec 18. 14]):
J. A WEAK FORMULATION OF DIFFUSIOJI\ PROBLEMS IN BROKEN SOBOLEV SPACES
We focus on a model linear diffusion problem in a bounded domain; given data (0., a0.. 51. f.g). we wish to lind a funct ion II such that
J. A WEAK FORMULATION OF DIFFUSIOJI\ PROBLEMS IN BROKEN SOBOLEV SPACES
where A E (L oc(0.»dxd is a diffusivity matrix satisfying the conditions stated in (2).
J. A WEAK FORMULATION OF DIFFUSIOJI\ PROBLEMS IN BROKEN SOBOLEV SPACES
The weak formulation of (\ X) that forms the basis of our discontinuous Galerkin method is defined on a broken space II (Pit). Pit being a member of a family of regular partitions of 0.. In particular. V(Ph) is a Hilbert space on the partition Pit. which is the completion of A complete characterization of the space Ii(Ph) for the one-dimensional case is described
J. A WEAK FORMULATION OF DIFFUSIOJI\ PROBLEMS IN BROKEN SOBOLEV SPACES
We should note that the terms h±a. with a = ]/2. are introduced to minimize the meshdependence of an otherwise strongly mesh-dependent nonn. The inner product associated with 11·11 v is the symmetric bilinear form
J. A WEAK FORMULATION OF DIFFUSIOJI\ PROBLEMS IN BROKEN SOBOLEV SPACES
.IrD
J. A WEAK FORMULATION OF DIFFUSIOJI\ PROBLEMS IN BROKEN SOBOLEV SPACES
[you])ds. (22) rlill Next, we introduce the bilinear form B±:
J. A WEAK FORMULATION OF DIFFUSIOJI\ PROBLEMS IN BROKEN SOBOLEV SPACES
• J'lfLI and the linear ronn L±: (24) rl,.EP;, Jrl,.
Jrll ./rN
In (23). we denote by (A Vv) . n} the limits of the sequcnces of averaged l1uxes (A Vvd . n} on rim. With these conventions and notations in place. we consider the following weak or variational boundary-value problem:
Jrll ./rN
Find U E V (Ph) such that (25) That (25) indeed corresponds to our model diffusion problem is veri tied in the following theorem:
Jrll ./rN
THEORE~13.1.
Suppose S, f. and g art! smooth (co/ltinl/ous) a/ld that the .I'o/Illion u to
Then II is a SO/III ion of (25) . COIlI'ersely, any slIfficiently smooth soilltio/l of (25) is also a so11lliO/l iif (i8).
Proof
This follo .... o'sfrom standard arguments and use of Green's romlUla. For details, see 114J .
Proof
• (26) the above inequality only indicates that the bilinear form is positive semiclellnite. 
The Global Element Method and Interior Penalty Formulations
The global elcment method [25. 261 consists in the classical hybrid fom1Ulation for a
The Global Element Method and Interior Penalty Formulations
Poissoll problem with the Lagrange multiplier eliminated in tet1l1S of the dependent variables: namely. the Lagrange multiplier is replaced by the average flux across interelement boundaries. The GEM can be written as follows:
The Global Element Method and Interior Penalty Formulations
A signilicant disadvantage of this method is that the matrix associated with the above bilinear form is indefinite (the real parts of the eigenvalues are not all positive), which prevents the solution of time-dependent di I'l'usion problems and also the utilization of many iterative schemes for thc solution of steady problems. The reason is lhat for advancing in timc thc solution to timc-dependent diffusion problems. the real part of the eigcnvalues of the space discretization needs to have the appropriate sign; otherwise the problem is unconditionally unstable.
The Global Element Method and Interior Penalty Formulations
Given that the goal of this investigation is to obtain a solver for convection-diffusion equations within the usual CFD settings. it is of paramount importance to generate a numerical technique whieh can handle these equations using pseudo-time-marching schemes.
The Global Element Method and Interior Penalty Formulations
The main disadvantage of the GEM is its inability to generate systems of equations which are amenable to the aforementioned solution techniques. Other than this disadvantage the GEM performs better (han the technique that \ve advocate when the error is measurcd in the L 2 norm (optimal for any p): this is not the case with the technique that wc advocatc. which loses one order for even powers p when the error is measured in the L 2 nonn (optimal for any p in the H t norm). The GEM has the advantage of producing symmetric systems of equations. but this advantagc is not of intcrest for convection-diffusion problems. which are intrinsically asymmetric (non-self-adjoint) problems. The variational formulation (25) will now be uscd as a basis for the construction of discontinuous Galcrkin approximations of the model diffusion problem. In the discontinuous Galerkinmcthod, the partitions of the solution domain. on which problem (18) is posed. arc now finite clements. across common boundaries of which the test functions can cxperiencc jumps. The Galerkin approximation is thus defined on a subspace VI'(P,,) of thc Hilbert space II (P,,) introduced in Section 2. Thus. the general setting of the approximation result (X) is applicable and can be used to derive £I priori error estimates for the method.
The Global Element Method and Interior Penalty Formulations
Let us now consider the finite-dimensional subspace Vp(P,,) C V (P,,) defined in Section 2.6 as
The Global Element Method and Interior Penalty Formulations
Vp(P,,) = IIPp,(O.,,).
,,=1
Our discontinuous Galerkin approximation (25) 
,,=1
where B+(·.·) and L+(·) are defined in (23) and (24) . rcspectively.
,,=1
An immediate observation is thar the discrete scheme dellncd by (30) is conservative. This is the subjcct of the following section.
L Strong Conservation at Element Level
A solution is said to be globally cOl/servative if the balance of the species (the solution II) is satisfied on the solution domain as a whole, and locally COI/.I'l'l'l'ative if a partition of thc solution domain exists such that thc balancc is salisllcd within each subdomain of this partition.
L Strong Conservation at Element Level
Considering a generic clement r2" E P", whcn the test function v" is a pieccwise constant (unit) function with local SllppOrl on clemem r2". from (30)- (23)- (24) we obtain the approximation
-l (AVII,,).nds-l.
,«(AVII1,)·n,,)ds= / Sdx+ r gds, (31) Im .. nrll ,m.,nl,,,, .In,.
-l (AVII,,).nds-l.
.Ian.nrN which means Ihat conservation at element level is ensured whcn the flux across iJr2" is defined as the average nux «(A VL'h)' n,.).
Numerical Evaluation of the Inf-Sup Condition
In order to apply the generalized Lax-Milgram theorem 15. 391 and the estimate (8). we must know the conrinuity and inf-sup parameters. The continuity condition holds with (19) , we obtain (33) The behavior of the discrete inf-sup parameter y" appearing in (8) as a function of mesh parameters hand p is of paramount importllnce in determining the stability of the DGM. A straightforward caleula! ion of y" for representative cases can be done using the eigenvalue problem described below.
Numerical Evaluation of the Inf-Sup Condition
THEORE1\'! 4.1. Let 1ih be an n-c/imemional Hilhert space, and CN" E 1R" X}t" the symmetric positive definite matrix associated 11'itltthe inner product in 1i". Let B: 1i" x 1i" --+ IR bc a gcncric bilinearjorm. amI BE jR" X IR" the associlltcd matrix: t!ten thc inlsllJl condition associated with B(-, .) cml hc cl'alullted usillg the cigcllrallll' prohlc/11 (34) /i'Olll which we obtaill the illl-wp constallt y" =J;'.mill' Proof Given that CN. is symmetric and positive definite. it can be factored into lower! upper triangular f0I111 eN" = u~"UN,.' Then. thc norm in 1i" is related to the Euclidenn using this result. we can write the discrete inf-sup constant y" as (35) and y" can be evaluated by solving the eigenvalue problem (34) .
Numerical Evaluation of the Inf-Sup Condition
•
Olle-Dill/{'/ISiolwJ Study
We consider thc simple Dirichlet problem -II" = f on (0, I to 8. For nonuniform meshes, both for the case of an arbitrary distribution of nodes in which the worst aspect ratio between adjacent elements is 114 and for a geometric distribution of mesh points such thaI hminl hmax = 10-7 , the values of y" are very close to those obtained with uniform meshes. the difference being of the order of 1%.
Olle-Dill/{'/ISiolwJ Study
In conclusion. the calculated y" is independent of the mesh size and is independelll of p for P888a:dbb7:2d49:b3f6:8605:bf4e:caed:e5a4: 3.
Olle-Dill/{'/ISiolwJ Study
The asymptotic values obtained are higher than the analytic value because the laller is only a lower bOllnd of the exact inf-sup constant.
Olle-Dill/{'/ISiolwJ Study
The a priori error estimate for this case is given by the following theorem: 
Two-Dimensional Analysis
A two-dimensional study of the behavior of the inf-sup parameter y" for the discrete case is carried out for meshes with different numbers N(P,,) of elements, different degrees of skewness, aspect ratios, and for uniform and nonuniform /le. These experiments are done for a model Dirichlet problems. for Laplace's e4uation on a parallelogram. Figure 5 shows some of the suhdomains on which the value of the inf-sup parameter y" is computed.
Two-Dimensional Analysis
First the inf-sup paramcter y" is evaluated for meshes with uniform p, p,. = p. 1 888a:dbb7:2d49:b3f6:8605:bf4e:caed:e5a4888a:dbb7:2d49:b3f6:8605:bf4e:caed:e5a4: e 888a:dbb7:2d49:b3f6:8605:bf4e:caed:e5a4888a:dbb7:2d49:b3f6:8605:bf4e:caed:e5a4: N (Ph). The numerical estimates indicate that the inf-sup parameter y" is asymptotically independcnt of 11, but depcnds on p. Figure 6 (top) shows the inf-sup dependence on /l for mcshes with M clements: uniform degree p: aspect ratios 1: 1, I :2, 1:4. 1:8: and skewness of 9lY and 30". The asymptotic dependence on p (within the range of intercst) is '" p-t for all the meshes with skewness of 90". and'" p-l.5 lor those with ske\\lness 0130".
Two-Dimensional Analysis
Next, the inf-sup parameter y" is evaluated for meshes with a random distribution of PC'
Two-Dimensional Analysis
The distribution starts wit h,)" = Pmax for element number 1, and for the remaining elements the order is chosen randomly between 2 and Pmax. The numerical estimates indicate that the inf-sup parameter y" is not very sensitive to abrupt changes in p. In fact, for onhogonal meshes. the inf-sup parameter y" is approximately 5% larger for most of the meshes where Mleast one element has Pr < Pmax. For skewed meshes in a few cases the inf-sup parametcr Yh is less than lO/C; smaller than the corresponding value when Pc = Pmax for allthe elements.
Two-Dimensional Analysis
and in many cases it is larger. From the above studies it is clear that the inf-sup parameter Yt, is asymptotically independent of 11 but depends on p. The dependence on p is not signilicant for practical calculations, since the loss of 0 (p) accuracy can be offset by the betler approximation achieved (0 (h P » using high p for cases with high regularity. By assuming that y" = O(p;;;~x)' K 2: O. an estimate of the global rates of convergence of the DGM can be easily estimated. We have: Ilu -u DeII~, 888a:dbb7:2d49:b3f6:8605:bf4e:caed:e5a4888a:dbb7:2d49:b3f6:8605:bf4e:caed:e5a4: c/I~~,X L:~-3!2-E 11/111, n..
Two-Dimensional Analysis
n.. E'P" pc (37) where l. and using the approximation estimate presented in (16) . there exists a local polynomial approximation /I p of /I E H S (Ph (S1») in the n0l11111· 11 v such that Finally. using the continuity and inf-sup parameters. and assuming that the exact solution " (P,,) , we arrive at the (/ priori error estimate where s > 3/2. /l" = min(p" + I. s), and C depends on s but is independent of /I. he. and Pc, • Remark 4. I. The error estimate (37) is a bound for the worst possiblc case. including all possible data. For a wide range of data, howevcr, the crror estimate (37) may be pessimistic. and the actual rate of convergence can be larger than that suggested by the ahove bound.
Two-Dimensional Analysis
The value of the parameter K depends on Pe and on d. For d = I, K = 0 regardless of Pe' as long as Pe 888a:dbb7:2d49:b3f6:8605:bf4e:caed:e5a4888a:dbb7:2d49:b3f6:8605:bf4e:caed:e5a4, whereas for d = 2 the value depends on the mesh regularity: numerical evidence suggests that for the cases considered K~1.0 -1.5, again for Pc 888a:dbb7:2d49:b3f6:8605:bf4e:caed:e5a4888a:dbb7:2d49:b3f6:8605:bf4e:caed:e5a4, as otherwise the method is unstable.
Nm:1ERICAL EXPERIMENTS
We shall now examine experimentally the perfollllance of the DGM for several representative examples.
Two-Point BV Problems
We will first analyze test cases of the Iype First we consider problem (39) with S = (471')2sin(4;rx), for which the exact solution is /lerae/(x) = sin(411'x). Figure 7 shows error in the nOllll 11·111/ and II convergence rate for uniform meshes. sin(2Jrx) on uniform meshes; Fig. 9 shows error in the L"-norm and h convergence rate. 
2-U Experiments
The lirsttest case is the solution to the Poisson problem
2-U Experiments
where n is the subdomain shown in Fig, 3 . Thc h convergence rate is evaluated by successive global refinements of the domain. Figure 12 shows the L2-norm of the error and the convergence rate. The second test case involves hop adaptation for a case with low regularity, which is a Dirichlet problem defined on the L-shaped domain shown in Fig. 4 . The above procedure to reline the mesh and to dccrease the order of the polynomial approximation is used until the values of the error indicators are below a prespe<.:itied value for every element in the mesh.
2-U Experiments
In the following test case the II-p adaptive process is initiated with a mesh consisting or three square elements of size I x I and polynomial basis of order 9. The purpose of this test is to show that the singularity at the corner is detected by the II-convergence rate information. Figure 13 shows the resulting p distribution and h refinemenl after live cycles. Note that the low regularity of the solution is detected hecause the order p is minimum (i.e .. p = 2 for stahi Iity reasons) for all the elements close to the corner singularity. Figure 14 shows a close-up view of pointwise error at thc corner.
2-U Experiments
In the following test we attempt to evaluate the sensitivity (loss of stability) of the method to p-enrichment in /.Ones with low regularity. In this case. the p distribution is obtained hy forcing p-enrichment where low regularity is detected (opposed to the usual procedure). Figure 16 shows fJ distrihution and II relinement. and Fig. 17 shows pointwise error. The convergence rate for this case is exponential. as shown in Fig. 18 , in the curve labeled hop sqn.
2-U Experiments
Finally. we compare convergence rates using uniform and adaptive refinements. Figure 18 shows the convergence rate using uniform p with global uniform relinements (curves labeled p = Il), and using lip-adaptation (labeled II-p com'). The convergence rate of un if 01111II refinements is exactly equal to the theoretical value N -1j.1 in the H I seminorm. and the convergence rates of the II-adaptation is close to the theoretical maximum which is N -I in the HI semi norm. The exponential convergence rates shown (as 11-p sqrt) are ohtained by setting the polynomial order as described hefore.
2-U Experiments
In summary, numerical experiments confirm the rohustness of the method under many different conditions. For the class of problems considered. the method appears to be stable even for arhitrary distributions of spectral orders and very di fl'erent element sizes and aspect ratios.
CONCLUDING COMMENTS
As a brief summary of the major observations of this study. we list the following:
CONCLUDING COMMENTS
• Diffusion dominated problems can be solved using piecewise discontinuous basis functions, without using auxiliary variahles such as fluxes in mixed methods. The discontinuous Galerkin method developed herein involves imposing weak continuity re4uirements on interclement houndaries: hoth solution values and fluxes are discontinuous across these intert~1ces.
CONCLUDING COMMENTS
• The method resembles hybrid and interior penalty methods. but no Lagrange multiplier or penalty parameter appears in the formulation.
CONCLUDING COMMENTS
• The method is robust. exhibiting only a small loss of accuracy in H t in which stretching and distortion of elements results in a suboptimal rate of p-convergence; numerical experiments suggest that for Pc 2: 2 the inf-sup parameter YI,~0 (p-l) for 2-D cases and
CONCLUDING COMMENTS
Yh is constant for I-D problems,
CONCLUDING COMMENTS
• The method is not stable for p"~I.
CONCLUDING COMMENTS
• The behavior of the method in L] is different for odd or even order polynomial approximations: for regular mesh relinements in 2-D. the L ]-rate of convergence is experimentally found to be 0 (111'+1) for p odd and 0 (hI') for p even, p 2: 2.
CONCLUDING COMMENTS
• The conditions under which the method is stable and convergent are studied herein. with corresponding a priori error estimates, and tests contiI'm that the method can exhibit high rates of 11-. p-, and lip-version convergence.
CONCLUDING COMMENTS
• Coupled with the classical discontinuous Galerkin formulation for transport dominated problems, this formulation is applicable to a wide range of problems, from convectiondominated to diffusion-dominated cases 1141.
CONCLUDING COMMENTS
• The formulation renders a numerical approximation which is elementwise conservative and. as such, is. to the best of our knowledge, the first high-order [inite element method ever developed with this property.
CONCLUDING COMMENTS
• The associated bilinear form renders a positive delinite and well-conditioned mutlix. thus allowing the use of standard iterative methods for high p and dist0l1ed elements.
CONCLUDING COMMENTS
• This formulation should be particularly convenient for time-dependent problems. because the global mass matrix is h!ock diagonal, with uncoupled blocks (see [14] ),

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