hp-Discontinuous Galerkin Finite Element Methods with Least-Squares Stabilization

3

0885-7474/02/1200-0003/0 © 2002 Plenum Publishing Corporation

Journal of Scientific Computing, Vol. 17, Nos. 1–4, December 2002 (© 2002)

hp-Discontinuous Galerkin Finite Element Methodswith Least-Squares Stabilization

Paul Houston,1 Max Jensen,2 and Endre Süli2

1 Department of Mathematics & Computer Science, University of Leicester, Leicester LE17RH, United Kingdom. E-mail: [email protected] Oxford University Computing Laboratory, Wolfson Building, Parks Road, Oxford OX1

3QD, United Kingdom. E-mail: {Max.Jensen;endre}@comlab.ox.ac.uk

Received August 8, 2001; accepted (in revised form) November 2, 2001

We consider a family of hp-version discontinuous Galerkin finite elementmethods with least-squares stabilization for symmetric systems of first-orderpartial differential equations. The family includes the classical discontinuousGalerkin finite element method, with and without streamline-diffusion sta-bilization, as well as the discontinuous version of the Galerkin least-squaresfinite element method. An hp-optimal error bound is derived in the associatedDG-norm. If the solution of the problem is elementwise analytic, an exponentialrate of convergence under p-refinement is proved. We perform numerical exper-iments both to illustrate the theoretical results and to compare the variousmethods within the family.

KEY WORDS: hp-finite element methods; discontinuous Galerkin methods;least-squares finite element methods; first order systems of PDEs.

1. INTRODUCTION

First-order systems of partial differential equations arise in a number ofphysical applications and their numerical solution is of great practicalinterest. Unfortunately, the standard Galerkin finite element approxima-tion of a first-order partial differential equation based on continuouspiecewise polynomials (CGFEM) exhibits poor stability properties whichinfrequently manifest themselves as large nonphysical numerical oscilla-tions whose frequency is commensurable with the mesh size.

Over the last two decades considerable progress has been made inthe area of stabilized finite element methods for the numerical solution offirst-order hyperbolic problems and second-order partial differential equa-tions with dominant hyperbolic behavior; a particularly significant method

from this class is the streamline-diffusion finite element method of Hughesand Brooks [15] which incorporates numerical dissipation in the charac-teristic (or streamwise) direction with the aim to suppress the undesirablenumerical oscillations in the standard continuous Galerkin finite elementmethod. A rigorous error analysis of this technique was performed byJohnson, Nävert and Pitkäranta [16]. More recently, the hp-version ofthe streamline-diffusion finite element method for first-order hyperbolicproblems and second-order partial differential equations with nonnegativecharacteristic form was considered by Houston, Schwab, and Süli [12] andHouston and Süli [14]; error bounds have been derived which, in thehyperbolic case, are simultaneously optimal with respect to the local meshsize h and the local polynomial degree p, with the local streamline-diffusionparameter d chosen to be of size h/p.

Another class of methods for first-order hyperbolic equations, basedon discontinuous piecewise polynomial approximation, was proposed in anearly paper of Reed and Hill [20]. The technique is commonly referredto in the literature as the discontinuous Galerkin finite element method(DGFEM). A rigorous error analysis of the method was carried out in aseries of papers by Lesaint and Raviart [18], Johnson and Pitkäranta [17],Richter [19] and Falk and Richter [10] for linear partial differential equa-tions. For nonlinear hyperbolic problems, the DGFEM was successfullyanalyzed and implemented by Cockburn and Shu [5–9]. The hp-version ofthe discontinuous Galerkin finite element method, with streamline-diffu-sion stabilization parameter d of size h/p2, was proposed and studied byBey and Oden [3]; they derived an h-optimal but p-suboptimal bound onthe global error of the method in the DG-norm. Recently, Houston, Schwaband Süli showed in [12, 22, 23] that with d of size h/p the method achievesan hp-optimal error bound in the DG-norm. It was also shown that ifthe solution to the underlying differential equation is elementwise analytic,then the method exhibits an exponential rate of convergence underp-refinement. Subsequently, by using a completely different error analysis,Houston, Schwab and Süli also proved in [13] that the hp-optimality ofthe method in the DG-norm persists even when the streamline-diffusionparameter is set to zero. This is in sharp contrast with the poor accuracyand stability properties exhibited by the CGFEM for first-order hyperbolicproblems in the absence of streamline-diffusion stabilization. The source ofthis difference between CGFEM and DGFEM is that the interelement jumpterms in DGFEM introduce at least as much numerical dissipation as stream-line-diffusion stabilization does, and therefore the inclusion of the latter intoDGFEM becomes unnecessary. An analogous remark applies when Galerkinleast-squares stabilization is used instead of streamline-diffusion stabilization.

While the streamline-diffusion finite element method (with continuouspiecewise polynomials) and the discontinuous Galerkin finite elementmethod (both with and without streamline-diffusion or least-squares stabi-lization) are stable and accurate numerical techniques, when applied to first-order systems of partial differential equations they lead to nonsymmetric

4 Houston, Jensen, and Süli

systems of linear algebraic equations, even if the system of partial differen-tial equations is symmetric. To rectify this, it is therefore appealing to con-sider least-squares finite element methods based on discontinuous piecewisepolynomials (LS-DGFEM). For a detailed survey of the theory of contin-uous least-squares finite element methods we refer to the review article ofBochev and Gunzburger [4]. It will be shown here that the hp-versionLS-DGFEM exhibits the same hp-optimal convergence rate in the asso-ciated DG-norm as the hp-version of the (continuous) streamline-diffusionfinite element method and the discontinuous Galerkin finite elementmethod (with or without streamline-diffusion or least-squares stabiliza-tion). LS-DGFEM leads to a system of linear algebraic equations with asymmetric positive definite matrix, thus enabling the use of standard androbust iterative algorithms such as a preconditioned conjugate gradientmethod. However, LS-DGFEM is not without disadvantages: even thoughits formal rate of convergence is identical to that of the streamline-diffusionfinite element method with continuous piecewise polynomials and thediscontinuous Galerkin finite element method with or without streamline-diffusion or least-squares stabilization, our numerical experiments indicatethat LS-DGFEM is more dissipative than either of the above methods.Indeed, the experiments reveal that the excess numerical dissipation inLS-DGFEM is due to the inclusion of least-squares stabilization into theinterelement jump terms rather than to the presence of least-squares sta-bilization in the element integral terms; this finding is consistent with theremark made at the end of the previous paragraph that the dominantsource of numerical dissipation in DGFEM are the interelement jumpterms. A further difficulty with LS-DGFEM is that the conditioning of the(symmetric) matrix for the associated system of linear equations is typicallyworse than that of the (unsymmetric) matrix arising from DGFEM. Thus,in order for LS-DGFEM to become competitive with DGFEM withrespect to the iterative solution of the resulting system of linear algebraicequations, it will be necessary to develop effective preconditioners for themethod; we shall not pursue this issue here any further other thanhighlighting its practical significance.

Despite LS-DGFEM’s inferiority in terms of practically observedaccuracy and the conditioning of the matrix of the resulting linear systemwhen compared with DGFEM, the combination of the two methods has ahelpful property: through the inclusion of least-squares stabilization intothe interelement jump terms the local numerical dissipation of the resultingmethod can be both increased and decreased beyond that of DGFEM; aswe have already indicated above, the amount of numerical dissipation inDGFEM cannot be controlled in a similar manner through the inclusionof streamline-diffusion stabilization as this has little or no effect on theresulting scheme. The chief purpose of this paper is to investigate thisparticular mix of DGFEM and LS-DGFEM through embedding the twomethods into a one-parameter family of discontinuous Galerkin finiteelement discretizations.

hp-Discontinuous Galerkin Finite Element Methods 5

The structure of the paper is as follows. In the next section we formulateour model problem: a symmetric system of first-order partial differentialequations. We then state the hp-version LS-DGFEM discretization of thismodel problem. In Section 3 we discuss the error analysis of the methodand prove that it exhibits hp-optimal convergence in the associated DG-norm; we also show that if the solution is elementwise analytic, then theconvergence rate of the method is exponential under p-refinement. InSection 4 we verify our theoretical results through numerical experiments;we also demonstrate that LS-DGFEM suffers from excessive numericaldissipation, and we identify least-squares stabilization in the interelementjump terms as the source of this undesirable feature. Thus, following theideas of Barth concerning least-squares stabilized discontinuous Galerkinfinite element methods ([1], [2]), and motivated by the findings of Sec-tion 4, we consider in Section 5 a general family of discontinuous Galerkinfinite element discretizations which admit completely irregular meshes withan arbitrary number of hanging nodes, and local and componentwisevariation of the polynomial degree in the finite element approximation. Thefamily includes DGFEM, with or without least-squares stabilization, aswell as LS-DGFEM. We show that the family exhibits hp-optimal conver-gence rates in the associated DG-norm; again, when the solution is ele-mentwise analytic the convergence rate is exponential under p-refinement.We conclude, in Section 6 with numerical experiments which demonstratethe performance of the various schemes.

2. MODEL PROBLEM AND THE hp-LS-DGFEM

Let W be a bounded open polyhedral domain in Rd, d \ 2, and let Cdenote the union of open faces of W. Suppose that B=(B1,..., Bd) is ad-component matrix function defined on W with Bi ¥ [W

1.(W)]

m×msymm , i=

1,..., d. Let m=(m1,..., md) denote the unit outward normal vector to C, andconsider the symmetric matrix B(m) — B ·m=m1B1+·· ·+mdBd. Supposethat B is diagonalized via B(m)=X(m)−1 L(m) X(m), whereL(m) is a diagonalmatrix, with the (real) eigenvalues of B(m) appearing along its diagonal. Weshall suppose that C is nowhere characteristic in the sense that none of thediagonal entries of L(m) is zero for any choice of the unit outward normalvector m on C. We decompose L(m) as L(m)=L−(m)+L+(m), where L−(m)is diagonal and negative semidefinite, and L+(m) is diagonal and positivesemidefinite. Thus we define

B−(m)=X(m)−1 L−(m) X(m) and B+(m)=X(m)−1 L+(m) X(m)

Clearly, for each m on C, B(m)=B−(m)+B+(m).Given C ¥ [L.(W)]m×m, f ¥ [L2(W)]m and g ¥ [L2(C)]m, we consider

the following boundary value problem: find u ¥H(L, W) such that

Lu — N · (Bu)+Cu=f in W, B−(m) u=B−(m) g on C (2.1)


where H(L, W)={v ¥ [L2(W)]m :Lv ¥ [L2(W)]m} denotes the graph spaceof the partial differential operator L in L2(W). Next, we formulate thehp-version of the least-squares Discontinuous Galerkin Finite ElementMethod (hp-LS-DGFEM, for short) for the numerical solution of (2.1).

Suppose that k \ 1, and Th is a regular or k-irregular subdivision of Winto disjoint open element domains o such that W=1o ¥Th

o. Thus a(d−1)-dimensional face of each element o in Th is allowed to contain atmost k hanging (irregular) nodes; when k=1, the hanging node is typicallythe barycenter of the face. We shall suppose that the family of subdivisionsTh is shape-regular and that each o ¥Th is a smooth bijective image of afixed master element o, that is, o=Fo(o) for all o ¥Th, where o is eitherthe open unit simplex or the open unit hypercube (−1, 1)d in Rd. On thereference element o, with x=(x1,..., xd) ¥ o and a=(a1,..., ad) ¥Nd

0 , wedefine spaces of polynomials of degree p \ 1 as follows:

Qp=span{xa : 0 [ ai [ p, 1 [ i [ d}, Pp=span{xa : 0 [ |a| [ p}

To each o ¥Th we assign an integer po \ 1; collecting the po and Fo in thevectors p={po : o ¥Th} and F={Fo : o ¥Th}, respectively, we introducethe finite element space

Sp(W,Th, F)={v ¥ [L2(W)]m : v|o p Fo ¥ [Qpo]m if F−1o (o)=(−1, 1)

d

and v|o p Fo ¥ [Ppo]m if F−1o (o) is the open unit simplex;

o ¥Th}

Given that Th is a subdivision of W, we define the broken Sobolev spaceH s(W,Th) of composite index s with nonnegative components so, o ¥Th, by

[H s(W,Th)]m={v ¥ [L2(W)]m : v|o ¥ [H so(o)]m -o ¥Th}

If so=s \ 0 for all o ¥Th, we shall simply write [H s(W,Th)]m.Given that o is an element in the subdivision Th, we denote by “o the

union of (d−1)-dimensional open faces of o. Let x ¥ “o and suppose thatno(x) denotes the unit outward normal vector to “o at x. With these con-ventions, we define B(no), B−(no) and B+(no) analogously to B(m), B−(m)and B+(m) above, respectively. For each o ¥Th and any v ¥ [H1(o)]m wedenote by v+o the interior trace of v on “o (the trace taken from within o).Now consider an element o such that the set “o0C is nonempty; then foreach x ¥ “o0C (with the exception of a set of (d−1)-dimensional measurezero) there exists a unique element o −, depending on the choice of x, suchthat x ¥ “o −. Suppose that v ¥ [H1(W,T)]m. If “o0C is nonempty for someo ¥Th, then we define the outer trace v−o of v on “o0C relative to o as theinner trace v+oŒ relative to those elements o − for which “o − has intersectionwith “o0C of positive (d−1)-dimensional measure. We also introduce thejump of v across “o0C: [v]o=v+o − v−o . Since below it will always be clearfrom the context which element o in the subdivision Th the quantities no,


v+o , v−o and [v]o correspond to, for the sake of notational simplicity weshall suppress the letter o in the subscript and write, respectively, n, v+, v−and [v] instead.

For v, w ¥ [H1(W,Th)]m, we introduce the bilinear form

BLS(w, v)= Co ¥Th

hopo

Fo

Lw ·MoLv dx

+ Co ¥Th

F“o0C

B−(n)[w] ·NoB−(n)[v] ds

+ Co ¥Th

F“o 5 C

B−(m) w+·NoB−(m) v+ ds (2.2)

Here, Mo ¥ Rm×msymm , No ¥ Rm×msymm , and Mo and No are bounded and positivedefinite, uniformly in o ¥Th, h and p.

For v ¥ [H1(W,Th)]m, we consider the linear functional

aLS(v)= Co ¥Th

hopo

Fo

f ·MoLv dx+ Co ¥Th

F“o 5 C

B−(n) g ·NoB−(n) v+ ds(2.3)

The hp-LS-DGFEM for (2.1) is defined as follows: find uLS ¥ Sp(W,Th, F)such that

BLS(uLS, v)=aLS(v) -v ¥ Sp(W,Th, F) (2.4)

The existence of a unique solution to (2.4) is easy to verify: as

BLS(w, w) > 0 -w ¥ Sp(W,Th, F)0{0}

the uniqueness of solutions to (2.4) follows; further, since (2.4) is a linearproblem over the finite-dimensional space Sp(W,Th, F), the existence of asolution is the consequence of its uniqueness.

3. ERROR ANALYSIS

Now, we embark on the error analysis of (2.4). We shall supposethroughout that u, the solution to our model problem (2.1), is sufficientlysmooth in the sense that u ¥ [H1(W,Th)]m and B−(n)[u]|“o=0 for anyo ¥Th (we note that a sufficient condition for this to hold is that u ¥[H1(W)]m). Under this hypothesis, we have the following Galerkinorthogonality property:

BLS(u−uLS, v)=0 -v ¥ Sp(W,Th, F)


We define the norm ||| · |||LS on [H1(W,Th)]m by |||w|||2LS=BLS(w, w). Thus,

|||u−uLS |||2LS=BLS(u−uLS, u−uLS)=BLS(u−uLS, u− v)

for all v ¥ Sp(W,Th, F) . Hence, on applying the Cauchy–Schwarz inequal-ity on the right-hand side,

|||u−uLS |||LS= infv ¥ Sp(W,Th, F)

|||u− v|||LS (3.5)

In the sequel, for the sake of simplicity, we shall suppose that W has beenpartitioned into elements o ¥Th each of which is a (bijective) affine imageof the reference hypercube o=(−1, 1)d. We recall from [12] the followingapproximation result; see also the monograph of Schwab [21], Theorem 3.17on p. 76.

Lemma 3.1. Suppose that u ¥ [Hko+1(o)]m for o ¥Th. Then, thereexists Pu ¥ Sp(W,Th, F) such that, for po \ 1 and 0 [ so [ min(po, ko), wehave

||u−Pu||2[L2(o)]m [ C 1ho222so+2 1

po(po+1)F(po, so) |u|

2[Hso+1(o)]m

and

|u−Pu|2[H1(o)]m [ C 1ho222so F(po, so) |u|2[Hso+1(o)]m

where

F(p, s)=(p−s)!(p+s)!

+1

p(p+1)(p−s+1)!(p+s−1)!

, 0 [ s [ p

and C is an absolute constant independent of ho, po and u.

Inserting these bounds into the multiplicative trace inequality

||w||2L2(“o) [ C(h−1o ||w||

2L2(o)+||w||L2(o) |w|H1(o)), w ¥H1(o)

we deduce the following error estimate.

Corollary 3.2. Under the hypotheses of Lemma 3.1,

||u−Pu||2[L2(“o)]m [ C 1ho222so+1 F1(po, so) |u|2[Hso+1(o)]m


where

F1(p, s)=1

`p(p+1)F(p, s)


In order to complete the error analysis we need to quantify |||u−Pu|||LS.This is easily achieved by substituting the last three error bounds into thedefinition of the norm ||| · |||LS.

Corollary 3.3. Under the hypotheses of Lemma 3.1 we have that

|||u−Pu|||2LS [ C Co ¥Th

1ho222so+1 LoF1(po, so) |u|2[Hso+1(o)]m

where

Lo=||Mo ||[L.(o)]m×m 1 ||B||2[L.(o)]m×m+> C+Cd

j=1

“Bj“xj>2[L.(o)]

m×m

2

+||No ||[L.(“o)]m×m ||B−(n)||2[L.(“o)]

m×m (3.6)


Our final bound on the discretization error is a trivial consequence of(3.5) and Corollary 3.3.

Theorem 3.4. Suppose that u ¥ [Hko+1(o)]m for each o ¥Th. Then,there exists a positive constant C, independent of ho, po and u such that, forpo \ 1 and 0 [ so [ min(po, ko), we have

|||u−uLS |||2LS [ C C

o ¥Th

1ho222so+1 LoF1(po, so) |u|2[Hso+1(o)]m

where Lo is defined by (3.6).

We conclude this section by an observation concerning the optimalityof the last error bound.

Remark 3.5. A simple application of Stirling’s formula shows that

F1(p, s)=C(s) p−(2s+1)

for all p \ 1. Hence, Theorem 3.4 gives

|||u−uLS |||2LS [ C C

o ¥Th

Lo 1hopo22so+1 |u|2[Hso+1(o)]m

with po \ 1 for all o ¥Th; this is optimal with respect to both ho and po.


Now, suppose that u is element-wise analytic on Th in the sense thatfor each o ¥Th the function u|o has analytic extension to an open set,independent of ho, containing o. Then,

,do > 0 ,C=C(u) > 0 -s \ 0 |u|[Hs(o)]m [ C(u)(do) s s! |meas(o)|12

By means of Stirling’s formula, after a rather straightforward but lengthycalculation, as in [12], we deduce from Theorem 3.4 the following result.

Theorem 3.6. Let W … Rd be a bounded polyhedral domain, Th={o} a shape-regular subdivision of W into d-parallelepipeds. Suppose that uis elementwise analytic on Th. Then, the solution uLS ¥ Sp(W,Th, F) of (2.4),with po \ 1 for o ¥Th, obeys the error bound

|||u−uLS |||2LS [ C(u) C

o ¥Th

1ho222so+1 p2oe−2bopo |meas(o)|

where C(u) is a constant depending on u, the dimension d, and the shape-regularity of Th; the real number bo is defined by

bo=12 |log F(amin, do)|

where

amin=(1+d2)−1/2 and F(a, d)=(1−a)1−a

(1+a)1+a(ad)2a

In the next section we shall verify the sharpness of our error bounds. Weshall also demonstrate that even though LS-DGFEM exhibits the sameasymptotic convergence rates as the classical DGFEM (cf. [12, 13]), it is notcompetitive with DGFEM as it suffers from excessive numerical dissipation.

4. NUMERICAL EXPERIMENTS

In this section we present a series of numerical experiments both toverify the a priori error estimates derived in Section 3 and to investigate thepractical performance of the hp-version LS-DGFEM.

4.1. Example 1

In this first example we consider a scalar advection-reaction problem (i.e.,m=1) with smooth analytical solution. To this end, we let W=(−1, 1)2,B(x)=(b1(x, y), b2(x, y)) — (8/10, 6/10), C(x)=c(x, y) — 1, g(x)=g(x, y)— 1 and f(x)=f(x, y) is chosen so that the analytical solution to (2.1) is

u(x) — u(x, y)=1+sin(p(1+x)(1+y)2/8) (4.7)

cf. [3, 12].


Fig. 1. Example 1. Convergence of the hp-LS-DGFEM with h-refinement.

Here, we investigate the asymptotic behavior of the hp-version of theleast-squares discontinuous Galerkin method on a sequence of successivelyfiner square and quadrilateral meshes for different p. In each case thequadrilateral mesh is constructed from a uniform N×N square mesh byrandomly perturbing each of the interior nodes by up to 10% of the localmesh-size, cf. [12].

In Fig. 1 we present a comparison of the least-squares norm (||| · |||LS) ofthe error with the mesh function h for 1 [ p [ 5. Here, we observe that|||u−uLS |||LS converges to zero, for each fixed p, at the rate O(hp+1/2) as themesh is refined, in agreement with Theorem 3.4. Finally, we investigatethe convergence of the hp-LS-DGFEM with p-enrichment on a fixed mesh.Since the true solution (4.7) is a real analytic function, we expect to seeexponential rates of convergence as p increases. In Fig. 2, we plot the least-squares norm of the error against p on four different square and quadrila-teral meshes. In each case, we observe that on a linear-log scale, theconvergence plots become straight lines as the spectral order p is increased,thereby indicating exponential convergence in p, cf. Theorem 3.6. Further-more, we observe from Figs. 1 and 2 that the h- and p-convergence of thehp-version LS-DGFEM are robust with respect to mesh distortion.

Fig. 2. Example 1. Convergence of the hp-LS-DGFEM with p-refinement.


4.2. Example 2Our second experiment compares the accuracy of LS-DGFEM with

that of DGFEM for a one-dimensional model problem with smooth solu-tion. We let W=(0, p) and select

B — R1 00 −1S , C — R0 1

1 0S , f — R0

0S , g(0)=−g(p)=R1

0S

The solution to this model problem is u=(cos(x), sin(x))T. Figure 3 indi-cates that both LS-DGFEM and DGFEM achieve optimal convergencerates in the L2-norm for this system. However, while we observe that forp \ 3 the errors of the two schemes are virtually identical, this is not so forp [ 2; indeed, in the latter case DGFEM delivers more accurate resultsthan LS-DGFEM on each of the uniform subdivisions of the interval (0, p)considered. This can also be seen in Fig. 4 which shows the L2-error againstthe polynomial degree.

In order to understand the observed behavior, let us consider asomewhat simpler scalar problem on W=(−1, 1), with Th={W}, B — 1and C — 0. Here the LS-DGFEM approximation can be expressed in closedform in terms of Legendre polynomials as

uLS(x)=Cp−2

i=0

2i+121F 1−1Li(t) u(t) dt2 Li(x)

+ Cp

i=p−1

121F 1−1Li−1(t) uŒ(t) dt2 Li(x)

Similarly, for DGFEM we obtain

uDG(x)=Cp−1

i=0

2i+121F 1−1Li(t) u(t) dt2 Li(x)

+1 C.

j=p

2j+12

F1

−1Lj(t) u(t) dt2 Lp(x)

Fig. 3. Example 2. Convergence of LS-DGFEM and DGFEM with h-refinement.


Fig. 4. Example 2. Convergence of LS-DGFEM and DGFEM with p-refinement.

To derive these identities we used integration by parts, the formula Ln=(L −n+1−L

−

n−1)/(2n+1), n \ 1, and the orthogonality of Legendre polyno-mials in the inner product of L2(−1, 1). Comparing uLS with uDG we seethat for p \ 3 the first p−2 terms in the expansions of uLS and uDG coin-cide; due to the fact that u is an entire analytic function on R, the higherorder Legendre modes decay very quickly and hence the difference betweenuLS and uDG is small for p \ 3, particularly for large p.

4.3. Example 3

In this third experiment we assess the practical performance of thehp-LS-DGFEM for a scalar linear advection problem with discontinuousboundary data. We let W=(0, 2)×(0, 1), B(x)=(b1(x, y), b2(x, y)) — (1+sin(py/2), 2), C(x)=c(x, y) — 0, f(x)=f(x, y) — 0 and

g(x) — g(x, y)=˛1, x=0, 0 [ y [ 1

sin6(px), 0 < x [ 1, y=0

0, 1 [ x [ 2, y=0

In Fig. 5, we compare the performance of the LS-DGFEM with thestandard discontinuous Galerkin method (DGFEM), and the streamline-diffusion stabilized discontinuous Galerkin method (SD-DGFEM). In eachcase, we show the outflow profile along the horizontal edge 0 [ x [ 2,y=1 on a 65×33 uniform square mesh with discontinuous piecewise bili-near elements (p=1). We observe that the performances of the DGFEMand the SD-DGFEM are very similar, cf. [12]; in each case the smooth hillis extremely well approximated, with some under-shoots and over-shootspresent in the vicinity of the discontinuity in the analytical solution. Incontrast, the LS-DGFEM is overly-diffusive leading to the excessivesmearing of both the discontinuity and the smooth hill present in the ana-lytical solution, cf. Fig. 5c. We remark that the numerical dissipationinherent in the LS-DGFEM is due to the inclusion of the least-squaresstabilization into the interelement jump terms rather than the presence of


Fig. 5. Example 3. Outflow profiles along 0 [ x [ 2, y=1, computed using DGFEMs on a65×33 mesh with piecewise bilinear elements (p=1): (a) Standard DGFEM; (b) Streamline-diffusion stabilized DGFEM; (c) LS-DGFEM; (d) Unsymmetric LS-DGFEM.

the least-squares stabilization in the element-integral terms. Indeed, inFig. 5d we show the solution generated by employing an unsymmetric least-squares discontinuous Galerkin finite element method where the elementintegral terms are identical to those in LS-DGFEM (cf. the first term in(2.2) and the first term in (2.3)) while the interelement jump terms and theboundary terms are identical to those arising in the standard DGFEM.Here, we observe that even though this unsymmetric least-squares methodis more dissipative than both DGFEM and SD-DGFEM, the excessivesmearing inherent in the LS-DGFEM has now been eradicated. Motivatedby these observations, in the following section we consider the analysis of ageneral class of discontinuous Galerkin schemes including DGFEM andLS-DGFEM as well as possibly new ones.

However, before concluding this section, we compare the performanceof the hp-LS-DGFEM with the standard, streamline-diffusion stabilizedand the Galerkin least-squares finite element methods based on continuouspiecewise polynomials. In Fig. 6 we show the outflow profiles of each ofthe aforementioned schemes; here, we again observe that the LS-CGFEMexcessively smears-out the solution, though the level of dissipation isslightly less than when the LS-DGFEM is employed.


Fig. 6. Example 3. Outflow profiles along 0 [ x [ 2, y=1, computed using continuousfinite element methods on a 65×33 mesh with piecewise bilinear elements (p=1): (a) Stan-dard CGFEM; (b) Streamline-diffusion stabilized CGFEM; (c) Least-squares CGFEM.

5. A GENERAL FAMILY OF DGFEMS

In this section, we embed LS-DGFEM into a one-parameter familyof discontinuous Galerkin finite element methods, which also includesDGFEM. We begin by recalling the DGFEM approximation of (2.1). Weadopt the following (standard) hypothesis: there exists a vector t ¥ Rd suchthat

12 (C(x)+CT(x))+12 (N ·B)(x)+B(x) ·t \ c0 Id a.e. x ¥ W (5.8)

where c0 > 0 is a constant. For simplicity of presentation, we shall assumethroughout that (5.8) may be satisfied with t — 0; we then define the posi-tive definite matrix C0 by

C0(x)=12 (C(x)+CT(x))+12 (N ·B)(x) a.e. x ¥ W (5.9)


Further, we let c1=maxx ¥ W ||C0(x)||2, where || · ||2 denotes the matrix norm onRm×m subordinate to the Euclidean norm on Rm. For v, w ¥ [H1(W,Th)]mwe consider the bilinear form

BDG(w, v)= Co ¥Th

Fo

w ·Lgv dx+ Co ¥Th

F“o 5 C

B+(m) w+· v+ ds

+ Co ¥Th

F“o0C

(B+(n) w++B−(n) w−) · v+ ds (5.10)

and the linear functional

aDG(v)= Co ¥Th

Fo

f · v dx− Co ¥Th

F“o 5 C

B−(m) gD · v+ ds (5.11)

The DGFEM approximation of (2.1) is defined as follows: find uDG ¥Sp(W,Th, F) such that

BDG(uDG, v)=aDG(v) -v ¥ Sp(W,Th, F) (5.12)

For a detailed stability and error analysis of this method in the case ofa scalar multi-dimensional hyperbolic problem we refer to the articles [12]and [13]. For the present purposes it is sufficient to note that

BDG(v, v)= Co ¥Th

Fo

C0v · v dx+12 Co ¥Th

F“o0C

−B−(n)[v] · [v] ds

+12 F“o 5 C

(B+(m)−B−(m)) v · v ds — |||v|||2DG (5.13)

Now suppose that t ¥ [0, 1]. Let us define the bilinear form

Bt(w, v)=(1−t) BLS(w, v)+tBDG(w, v)

and the linear functional

at(v)=(1−t) aLS(v)+taDG(v)

We consider the following approximation of (2.1) which we shall hence-forth refer to as tLS-DGFEM: find uhp ¥ Sp(W,Th, F) such that

Bt(uhp, v)=at(v) -v ¥ Sp(W,Th, F) (5.14)

Clearly,

Bt(v, v)=(1−t) |||v|||2LS+t |||v|||

2DG > 0 -v ¥ Sp(W,Th, F)0{0} (5.15)

Consequently, (5.14) has a unique solution uhp ¥ Sp(W,Th, F).


Next we turn our attention to the error analysis of tLS-DGFEM. Asthe case of t=0 has already been dealt with in Section 2, and hp-DGFEM,corresponding to t=1 has been shown to be hp-optimally convergent in[13], we shall confine our analysis to the case when t ¥ (0, 1).

First, we shall weaken the hypotheses of Section 3 on Mo and No bysupposing instead that Mo ¥ Rm×msymm and No ¥ Rm×msymm are bounded, uniformlyin o ¥Th, h and p, and there exists a constant m0 ¥ (0, t), independent of hand p such that, for all o in Th,

˛Mo \ m0Id and(−B−)(n){2(1−t) No(−B−)(n)+(t−m0) Id} \ 0 on “o

(5.16)

For example, the second condition in (5.16) is satisfied with No — 0 for allo ¥Th; indeed, No need not even be positive semidefinite; it suffices that

(−B−)(n) No(−B−)(n) \ − y (−B−)(n) , where y=t−m02(1−t)

(5.17)

This means that, locally, the level of numerical dissipation in tLS-DGFEMcan be reduced or increased beyond that of the basic DGFEM, which is ahelpful computational device.

We begin the error analysis of tLS-DGFEM by observing that ifu ¥ [H1(W,Th)]m and B−(n)[u]=0 for all o ¥Th, then

Bt(u−uhp, v)=0 -v ¥ Sp(W,Th, F) (5.18)

On decomposing u−uhp=t+g where t=Pu−uhp and g=u−Pu, withPu defined as in Section 3, we conclude from (5.18) that

Bt(t, t)=−Bt(g, t)=−(1−t) BLS(g, t)−tBDG(g, t)

— T1+T2 (5.19)

We begin by estimating the left-hand side of (5.19) from below. We deducefrom (5.13) and the definition of BLS( · , · ) that

Bt(t, t)= Co ¥Th

1 t Fo

t ·C0t dx+(1−t) Fo

hopo

Lt ·MoLt dx2

+12

Co ¥Th

F“o0C

(−B−(n))[t] · { t Id+2(1−t) No(−B−(n))}[t] ds

+12

Co ¥Th

F“o 5 C

(−B−(m)) t+· { t Id+2(1−t) No(−B−(m))} t+ ds

+t2

Co ¥Th

F“o 5 C

B+(m) t+·t+ ds


Hence,

Bt(t, t) \ tc0 Co ¥Th

||t||2L2(o)+(1−t) m0 Co ¥Th

hopo||Lt||2L2(o)

+12m0 C

o ¥Th

||(−B−(n))12 [t]||2L2(“o0C)

+12m0 C

o ¥Th

||(−B−(m))12 t+||2L2(“o 5 C)

+12t Co ¥Th

||(B+(m))12 t+||2L2(“o 5 C) (5.20)

with 0 < m0 < t < 1 and c0 > 0.Next, we bound T1 from above. Recall that Mo and No have been

assumed uniformly bounded, independent of ho and po; therefore, thereexists a constant m1 > 0, independent of ho and po such that

Mo [ m1Id and |No | [ m1Id for all o ¥Th

Further, as Bi ¥ [W1.(W)]

m×msymm for i=1,..., m, there exists a positive con-

stant b1 such that, with n=no,

max(||−B− (n)(x)||2 , ||B+(n)(x)||2) [ b1 -o ¥Th and all x ¥ “o0C

and, with m(x)=mo(x) for x ¥ “o 5 C,

max(||−B− (m)(x)||2 , ||B+(m)(x)||2) [ b1 -o ¥Th and all x ¥ “o 5 C

where, again, || · ||2 denotes the matrix norm on Rm×m subordinate to theEuclidean norm on Rm. Thus,

T1 [ (1−t) m1 1 Co ¥Th

hopo||Lg||2L2(o) 2

12 1 C

o ¥Th

hopo||Lt||2L2(o) 2

12

+(1−t) m1b1 1 Co ¥Th

||(−B−(n))12 [g]||2L2(“o0C) 2

12

×1 Co ¥Th

||(−B−(n))12 [t]||2L2(“o0C) 2

12

+(1−t) m1b1 1 Co ¥Th

||(−B−(m))12 g+||2L2(“o 5 C) 2

12

×1 Co ¥Th

||(−B−(m))12 t+||2L2(“o 5 C) 2

12

(5.21)


Next we bound T2 from above. To do so, we begin by noting that

Co ¥Th

F“o0C

B+(n) g+·t+ ds=− Co ¥Th

F“o0C

B−(n) g− ·t− ds

and therefore,

: Co ¥Th

F“o0C

(B+(n) g++B−(n) g−) t+ ds :

[ 1 Co ¥Th

||(−B−(n))12 g−||2L2(“o0C) 2

12 1 C

o ¥Th

||(−B−(n))12 [t]||2L2(“o0C) 2

12

Hence, noting that Lgt=−Lt+2C0t, we deduce that

T2 [ t 1 Co ¥Th

poho||g||2L2(o) 2

12 1 C

o ¥Th

hopo||Lt||2L2(o) 2

12

+2tc1 1 Co ¥Th

||g||2L2(o) 212 1 C

o ¥Th

||t||2L2(o) 212

+t 1 Co ¥Th

||(−B−(n))12 g−||2L2(“o0C) 2

12 1 C

o ¥Th

||(−B−(n))12 [t]||2L2(“o0C) 2

12

+t 1 Co ¥Th

||(B+(m))12 g+||2L2(“o 5 C) 2

12 1 C

o ¥Th

||(B+(m))12 t+||2L2(“o 5 C) 2

12

(5.22)

Upon inserting the upper bounds (5.21) and (5.22) on T1 and T2,respectively, and the lower bound (5.20) on Bt(t, t) into (5.19), we findthat

|||t|||2t — Co ¥Th

||t||2L2(o)+ Co ¥Th

hopo||Lt||2L2(o)+ C

o ¥Th

||(−B−(n))12 [t]||2L2(“o0C)

+ Co ¥Th

||(−B−(m))12 t+||2L2(“o 5 C)+ C

o ¥Th

||(B+(m))12 t+||2L2(“o 5 C)

[ y1 |||g|||gt |||t|||t (5.23)


where y1=y1(t, c0, c1, m0, m1, b1) is a positive constant, independent of thediscretization parameters, and

|||g|||gt=1 Co ¥Th

3poho||g||2L2(o)+

hopo||Lg||2L2(o)+||(−B−(n))

12 [g]||2L2(“o0C)

+||(−B−(n))12 g−||2L2(“o0C)+||(−B−(m))

12 g+||2L2(“o 5 C)

+||(B+(m))12 g+||2L2(“o 5 C) 42

12

We thus conclude from (5.23) that

|||u−uhp |||t [ |||t|||t+|||g|||t [ y1 |||g|||gt+|||g|||t

so to complete the error analysis we now only need to bound |||g|||gt and|||g|||t in exactly the same manner as in Section 3. Thus we arrive at thefollowing result.

Theorem 5.1. Suppose that (5.16) holds and that u ¥ [Hko+1(o)]mfor each o ¥Th. Then, for each t ¥ (0, 1) there exists a positive constant C,independent of ho, po and u, such that, for po \ 1 and 0 [ so [ min(po, ko),we have

|||u−uhp |||2t [ C C

o ¥Th

1ho222so+1 F1(po, so) |u|2[Hso+1(o)]m

Remark 5.2. A simple application of Stirling’s formula shows that

F1(p, s)=C(s) p−(2s+1)

for all p \ 1. Hence Theorem 5.1 gives

|||u−uhp |||2t [ C C

o ¥Th

1 hopo22so+1 |u|2[Hso+1(o)]m

with po \ 1, 0 [ so [ min(po, ko), for all o ¥Th, where C is a positive con-stant independent of ho, po and u. This bound is optimal with respect toboth ho and po.

Now, suppose that u is element-wise analytic on Th in the sense thatfor each o ¥Th the function u|o has analytic extension to an open set,independent of ho, containing o. Then, analogously as in Section 3, wededuce from Theorem 5.1 the following result.

Theorem 5.3. Suppose that (5.16) holds and let W … Rd be abounded polyhedral domain, Th={o} a shape-regular subdivision of Winto d-parallelepipeds. Assume that u is elementwise analytic on Th. Then,


for any t ¥ (0, 1), the solution uhp ¥ Sp(W,Th, F) of (5.14), with po \ 1 foro ¥Th, obeys the error bound

|||u−uhp |||2t [ C(u) C

o ¥Th

1ho222so+1 p2oe−2bopo |meas(o)|

where C(u) is a constant depending on u, the dimension d, and the shape-regularity of Th; the real number bo is defined by bo=|log F(amin, do)|/2,where

amin=(1+d2)−1/2 and F(a, d)=(1−a)1−a

(1+a)1+a(ad)2a

6. NUMERICAL EXPERIMENTS

In this section we shall explore the implementation of tLS-DGFEMand conduct numerical experiments to verify the sharpness of our analyticalresults.

6.1. Example 4

We consider the one-dimensional wave equation

“2f

“t2−c2“2f

“x2=0 (6.24)

We note that by defining x1 — x and x2 — t, the second-order partial dif-ferential equation (6.24) can be rewritten as the symmetric first-ordersystem

r 0 −c−c 0s ru1u2sx1

+r1 00 1s ru1u2sx2

=r00s (6.25)

where u1=“f/“x2 and u2=c“f/“x1. Here we let W=(0, 1)×(0, 1/2) andc=1/2, with

u1(x1, 0)=0 and u2(x1, 0)=ce−100(x1 −1/2)2

for 0 [ x1 [ 1; this is a variant of the test problem considered in [11].In Fig. 7 we compare the performance of the tLS-DGFEM with t=0

(LS-DGFEM), t=1/2 and t=1 (DGFEM), using h-refinement onuniform square meshes for 1 [ p [ 5. For consistency, in each case theerror is measured in terms of the L2(W) norm. Here, we observe that theerror in the DGFEM is always smaller than for the tLS-DGFEM witht=1/2; though the error for this latter scheme is always smaller than forthe LS-DGFEM. However, if we measure the error with respect to the


Fig. 7. Example 4. Convergence of the tLS-DGFEM with h-refinement.

L.(W) norm, then we see that the tLS-DGFEM with t=1/2 now has clearadvantages over both the DGFEM and the LS-DGFEM. Indeed, fromFig. 8, we see that for p > 1, the tLS-DGFEM with t=1/2 now outper-forms both DGFEM and LS-DGFEM on some of the meshes employed;this is particularly noticeable on coarser grids. We note that the differencesbetween these three schemes are quite small for this smooth problem;however, motivated by our findings here, in the following section we con-sider a model problem with a nonsmooth analytical solution.

6.2. Example 5

We end this paper by investigating the performance of the tLS-DGFEM on the (nonsmooth) linear advection problem considered inSection 4.3. To this end, in Fig. 9 we plot the L.(W) norm of the error inthe tLS-DGFEM for 0 [ t [ 1 on a 33×17 uniform square mesh for1 [ p [ 6. Here, we observe that as t increases from t=0 (corresponding tothe LS-DGFEM), the infinity norm of the error first decreases, beforeincreasing sharply as t approaches one (corresponding to the DGFEM).This implies that for each p there is an optimal value of the parametert=tp, for which the infinity norm of the error in the tLS-DGFEM isminimized. We remark that the error curves for each p are fairly flat

Fig. 8. Example 4. Convergence of the tLS-DGFEM with h-refinement.


Fig. 9. Example 5. Comparison of the L.(W) norm of the error with t on a 33×17 uniformsquare mesh.

around this hypothetical value tp, which is good from the practical point ofview, since there is a fairly large range of t around tp which gives roughlythe same error as tp itself.

Thereby, we see that the dissipation present in the tLS-DGFEM, witht % tp, is large enough to provide additional stabilization for higher p, cf.[12], yet small enough to ensure that the error is not adversely affected bythe excessive dissipation present in the LS-DGFEM, cf. Section 4.3. Ofcourse, the level of dissipation added into the standard DGFEM by com-bining it with the LS-DGFEM can be controlled not only by varying t, butalso by allowing Mo and No, cf. (2.2), to change from element to element inthe mesh Th. Indeed, for a fixed t, Mo and No could be selected based onthe size of the finite element residual to ensure the necessary ‘‘tune-up’’ ofthe numerical dissipation in the vicinity of sharp features in the analyticalsolution u; this work is part of our current research programme and will bepresented elsewhere. In the present experiments we simply chose Mo=No=Id for all o ¥Th.

ACKNOWLEDGMENTS

Paul Houston acknowledges the financial support of the EPSRC(Grant GR/N24230). Max Jensen thanks the DAAD (‘‘Doktorandensti-pendium im Rahmen des gemeinsamen Hochschulprogramms III von Bundund Ländern’’), the Gottlieb Daimler- und Karl Benz-Stiftung and theFlughafen Frankfurt Main Stiftung for their sponsorship.

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