Ainsworth Optimally Blended Spectral Fe Scheme for Wave Propagation and Nonstandard Reduced Integration

7/31/2019 Ainsworth Optimally Blended Spectral Fe Scheme for Wave Propagation and Nonstandard Reduced Integration

1/26

SIAM J. N UMER. A NAL . c 2010 Society for Industrial and Applied MathematicsVol. 48, No. 1, pp. 346371

OPTIMALLY BLENDED SPECTRAL-FINITE ELEMENT SCHEMEFOR WAVE PROPAGATION AND NONSTANDARD REDUCED

INTEGRATION

MARK AINSWORTH AND HAFIZ ABDUL WAJID

Abstract. We study the dispersion and dissipation of the numerical scheme obtained by takinga weighted averaging of the consistent (nite element) mass matrix and lumped (spectral element)mass matrix for the small wave number limit. We nd and prove that for the optimum blending theresulting scheme (a) provides 2 p + 4 order accuracy for pth order method (two orders more accuratecompared with nite and spectral element schemes); (b) has an absolute accuracy which is O ( p 3 )and O ( p 2 ) times better than that of the pure nite and spectral element schemes, respectively; (c)tends to exhibit phase lag. Moreover, we show that the optimally blended scheme can be efficientlyimplemented merely by replacing the usual Gaussian quadrature rule used to assemble the mass andstiffness matrices by novel nonstandard quadrature rules which are also derived.

Key words. numerical dispersion, numerical dissipation, high order numerical wave propagation

AMS subject classications. 65N15, 65N30, 65N35, 35J05

DOI. 10.1137/090754017

1. Introduction. The development and analysis of numerical methods for wavepropagation frequently centers on the issue of controlling errors due to numericaldispersion and dissipation [4, 8]. The dispersive properties of standard Galerkin nite

element methods of arbitrary order were considered in [1]. However, spectral elementmethods [9, 10, 11, 12] have attracted considerable interest in the computational wavepropagation community thanks in part to the fact that the mass matrix is diagonal,and their superior phase accuracy compared with standard nite elements [2].

Even as early as 1984, the possibility of employing a weighted average of thenite element and spectral element schemes has been conjectured as a means bywhich to obtain the most promising, cost-effective method for computational wavepropagation [13]. Many authors have even commented on the effectiveness of thescheme obtained by forming a simple average of the spectral and nite element schemesin the case of rst order elements, but no systematic treatment or analysis seems to

be available. Seriani and Oliveira [15] consider the possibility of blending the methodsusing a criterion whereby the phase error vanishes at a particular, user-specied, valueof the normalized wave number. However, this approach means that the blendingparameter is frequency and mesh dependent and may actually result in an increasein the phase error at frequencies that were originally resolved by the pure nite andspectral element approaches.

A more natural approach to the selection of the blending parameter that is more inthe spirit of the design of methods for computational wave propagation is to maximizethe order of accuracy in the phase error. This criterion is adopted in the present work.

Received by the editors March 26, 2009; accepted for publication (in revised form) December 3,2009; published electronically April 16, 2010.

http://www.siam.org/journals/sinum/48-1/75401.html Mathematics Department, University of Strathclyde, 26 Richmond Street, Glasgow G1 1XH,

Scotland ([email protected], [email protected]). The rst author was supportedby the Engineering and Physical Sciences Research Council under grant EP/E040993/1. The sec-ond author was supported by COMSATS Institute of Information Technology, Pakistan through aresearch studentship which is gratefully acknowledged.

346


2/26

OPTIMALLY BLENDED SCHEME FOR WAVES 347

We show that the optimal choice of blending parameter for elements of order p Nis given by weighting the spectral element to the nite element in the ratio p : 1. Arigorous proof of this fact is provided along with precise error estimates and ordersof accuracy in the phase error. In particular, we show that the optimally blendedscheme gives an additional two orders of accuracy compared with the pure schemes.

An ostensibly different approach to the construction of nite element-like schemesfor wave propagation consists of using nonstandard quadrature rules. The thesis of Challa [3] presents nonstandard quadrature rules for both linear and quadratic niteelements that lead to an improvement in the order of accuracy of the phase error.Subsequently, Guddati and Yue [6, 7] studied such schemes for linear nite elementsand commented on the relation with blended spectral-nite element schemes in thecase of rst order elements.

The nonstandard quadrature rule provides an attractive means by which to im-

plement the blended spectral-nite element approach in the case p = 1. Accordingly,we investigate whether or not suitable nonstandard quadrature rules exist in the caseof elements of arbitrary order p N , such that the resulting scheme is identical tothe optimally blended spectral-nite element method described above. We show suchnonstandard quadrature rules exist for all orders, give an explicit construction for theweights and nodes, and study their properties.

In summary, we believe that the present work conrms the earlier remark made byMarfurt [13] and provides a simple means by which optimally blended spectral-niteelement schemes can be efficiently realized in practice.

2. Motivation and overview of main ideas and results. In order to moti-vate the ideas, we begin by presenting the discrete dispersion analysis of the simple1D model problem

(2.1) 2U t 2

2U x 2

= 0 , x R , t > 0,

along with appropriate initial and boundary conditions. Our chief interest in thepresent work is the study of the impact of the spatial discretization on the phaseaccuracy, and we therefore consider a time-harmonic solution of the form U (x, t ) =e it u(x) for a given temporal frequency R . This leads us to consider the spatialdiscretization of the model problem:

(2.2) u (x) 2u(x) = 0 , x R .2.1. Piecewise linear approximation in one dimension. Suppose we dis-

cretize (2.2) using piecewise linear nite elements on a uniform mesh of size h > 0and seek an approximation of the form

j Zu j j (x),

where j is the usual piecewise linear hat function associated with node x j . We obtainthe following equation for the value u j of the approximation at node x j = jh, j Z :

(2.3) u j +1 2u j + u j 1 +26

(u j +1 + 4 u j + u j 1) = 0 ,

where = h. This equation admits nontrivial solutions of the form u j = eij(1) h

provided that (1) h satises

(1) h = cos 16 226 + 2


3/26

348 MARK AINSWORTH AND HAFIZ ABDUL WAJID

or, writing the above expression as a series in ,

(2.4) (1) h = 3

24+ .

This result shows that the nite element approximation tends to exhibit phase lead compared with the corresponding solutions of the continuous equation (2.2).

An alternative approach is to approximate the problem using the spectral elementmethod. This results in the following equation for the coefficients u j :

(2.5) u j +1 2u j + u j 1 + 2 u j = 0 ,leading to nontrivial solutions of the form u j = eij

(1) h , where

(1)

h = cos 1

1 2

2

or, again expanding as a series in ,

(2.6) (1) h = +3

24+ .

This means that the spectral element scheme tends to exhibit phase lag . In search of a numerical scheme with superior phase accuracy, we follow the suggestion of Marfurt[13] and form a blended scheme by taking a linear combination of (2.3) and (2.5):

(2.7) u j +1 2u j + u j 1 + 2

6(1 )u j +1 + 2(2 + )u j + (1 )u j 1 = 0 ,

where [0, 1] is a parameter whose value is to be determined.Proceeding as before, we discover that the scheme admits nontrivial solutions of

the form u j = eij(1) h , where (1) depends on , and is given by

(2.8) (1) h = cos 1 2(2 + ) 6

2( 1) 6,

or, writing the above expression as a series in ,

(2.9) (1) h = +3

24(2 1) +

5

1920(20 2 20 + 9) + .

The above expression reduces to those obtained for the nite and spectral elementschemes in the cases = 0 and = 1, respectively. However, more interestingly, weobserve that by choosing = 1 / 2, two additional orders of accuracy in the phase areobtained.

2.2. Implementation via nonstandard quadrature rules. The practicalimplementation of the blended scheme may at rst sight appear to entail the assemblyof the mass matrices for both the nite and spectral element schemes, which wouldbe rather unattractive. We can construct another piecewise linear nite elementapproximation in which the entries in the mass and stiffness matrices are approximatedusing the nonstandard quadrature rule

(2.10) 1 1 f (x)dx Q(1) (f ) = f () + f () ,


4/26


where = 13 (1 + 2 ). This rule is exact for linear functions, but not products of linear functions, meaning that the entries appearing in the mass matrix are under-integrated. The quadrature rule (2.10) is used to develop a composite quadrature rule

I (1),h on R given by

R f (x)dx h2 j Z f (+j ) + f (j ) = I (1),h (f ),where j = ( j +

12 )h h2 , j Z .The new piecewise linear nite element approximation is then dened by seeking

a nontrivial function of the form

U h (; x) =j Z

u j j (x), x R ,

such that

(2.11) I (1),h ( x U

h x r ) 2 I

(1),h (U

h r ) = 0

for all r Z . Interestingly, the resulting scheme gives precisely the same stencil as(2.7) for the coefficients u j j Z , where, as before, = h:

u j +1 2u j + u j 1 +2

6(1 )u j +1 + 2(2 + )u j + (1 )u j 1 = 0 .

In other words, the scheme coincides with the blended scheme in the case of linearelements, meaning that the blended scheme can be realized in practice by replacingthe standard Gaussian quadrature rule by the nonstandard rule (2.10). Similarly,the optimally blended scheme can be obtained by using the quadrature rule (2.10) inconjunction with the choice = 1 / 2.

In summary, the nonstandard quadrature rule leads to a scheme which admits anontrivial solution given by

(2.12) U h (; x) =j Z

eij(1) h j (x),

where (1) is dened in (2.8) with = h.

2.3. Extension to multiple spatial dimensions. We now turn to the caseof higher dimensional problems and investigate whether the blending of spectral andnite element approximation offers similar advantages to those observed in one spatialdimension. Suppose we discretize the equation

(2.13) u 2u = 0 in R 3using a tensor product grid hZ 3 on R 3 , in conjunction with trilinear basis functions.

A standard nite element implementation requires the use of a quadrature rule toapproximate the integrals over the reference element K = [

1, 1]3 . Generally, a tensor

product GaussLegendre rule would be applied. However, prompted by the earlierobservation we propose to use instead a tensor product rule based on the nonstandardquadrature rule (2.10):

K f (x,y,z )dxdydz f ( , ,) + f (,, ) + f (, , ) + f ( , ,)+ f (, , ) + f (,, ) + f (, , ) + f (, , ),(2.14)


5/26


where = 13 (1 + 2 ). If we choose = 0, then the scheme reduces to the standardnite element approximation while the choice = 1 gives the spectral element scheme.Consequently, the scheme with a general choice of may be considered as a blended

approximation. We wish to analyze the dispersive properties of the resulting scheme.Based on our experience in the one dimensional case, we seek a nontrivial solution inthe three dimensional case in the form

u(x,y,z ) = U h (kx ; x)U h (ky ; y)U

h (kz ; z),

where kx , ky , kz R are constants to be determined and U h is dened in (2.12).Observe that, thanks to (2.12), we could also write

u(x,y,z ) =,m,n

eih [ (1) (k x )+ m

(1) (k y )+ n

(1) (k z )] (x)m (y)n (z),

where (1) (k) is given by (2.8) with = h, which, as one would expect, indicatesthat u corresponds to a plane wave. Inserting this expression into the approximatebilinear form associated with the quadrature rule (2.14) and using a test functionv(x,y,z ) = q(x)r (y)s (z) leads to

I (1),h ( x U

h (kx ; x) x q) I

(1),h (U

h (ky ; y) r ) I

(1),h (U

h (kz ; z) s )

+

I

(1),h (U

h (kx ; x) q)

I

(1),h ( y U

h (ky ; y) y r )

I

(1),h (U

h (kz ; z) s )

+ I (1),h (U

h (kx ; x) q) I

(1),h (U

h (ky ; y) r ) I

(1),h ( z U

h (kz ; z) z s )

= 2 I (1),h (U

h (kx ; x) q) I

(1),h (U

h (ky ; y) r ) I

(1),h (U

h (kz ; z) s )(2.15)

for all q,r,s Z . Recalling that U h satises (2.11) leads to the following conditionfor the parameters kx , ky , and kz :

(k2x + k2y + k

2z 2) I

(1),h (U

h (kx ; x) q) I

(1),h (U

h (ky ; y) r ) I

(1),h (U

h (kz ; z) s ) = 0

and as a consequence, we deduce that the new scheme admits a nontrivial solutionprovided that

k2x + k2y + k

2z =

2 .

The wave number of the discrete solution is given by kh , where

k2h = (1) (kx )

2 + (1) (ky )2 + (1) (kz )

2 ,

and then, thanks to (2.9), we deduce that

k2h = 2 +

h212

(2 1)[(kx )4 + ( ky )4 + ( kz )4] + O(h46).We again see that there exists an optimal choice of blending parameter, and moreover,it coincides with the optimal parameter for the one dimensional case. The argumentsused above extend to any number of dimensions meaning that the optimal blendingparameter is independent of the number of spatial dimensions.


6/26


2.4. Numerical example. In practice, by making use of the nonstandardquadrature rule, the cost of using the optimally blended scheme is virtually the sameas that of using the pure nite or spectral element scheme, but can result in markedlysuperior numerical results. In order to illustrate the potential of such an approach inmultidimensions, we consider the problem

(2.16) 2u(x, y ) 2u(x, y ) = 0 in (0 , 1)2subject to Dirichlet boundary conditions:

u = eik 1 x on 1 = {x (0, 1), y = 0},u = eik 2 y on 4 = {y (0, 1), x = 0},

where k1 and k2 are user-specied constants satisfying k21 + k22 = 2 , and nonreectingboundary conditions:

ux Gu = 0 on 2 = {y (0, 1), x = 1 },uy Gu = 0 on 3 = {x (0, 1), y = 1},

where G is the usual Dirichlet to Neumann map [8]. The Dirichlet boundary conditionsare chosen so that the exact solution to the boundary value problem (2.16) is the planewave solution u(x, y ) = ei (k 1 x + k2 y ) , with the coefficients chosen to be k1 = 20 andk2 = 1. The accuracy of the real components of the spectral, nite, and optimal

scheme solutions obtained with 20 linear elements are compared in Figure 2.1, wherethe cuts of the two dimensional solution along the lines y = x and y = 2 x relativeto the edge 1 are shown. The phase lead and lag are evident and correspond tonumerical approximations obtained using the nite and spectral element schemes,respectively. Moreover, the phase accuracy of the numerical approximation obtainedusing the optimal scheme is noticeably better than that of nite and spectral elementschemes. In Figure 2.2, we show the effect of increasing the number of elements ineach direction along the same lines as used in Figure 2.1. It is clear that with 30linear elements the numerical approximations obtained using the nite and spectralelement schemes converge to the exact solution but phase lead and phase lag are still

prominent, while the numerical approximation corresponding to the optimal schemeis virtually completely resolved.

2.5. Extension to quadratic elements. In the case of quadratic elements, weobtain two discrete equations corresponding to the nodal and midpoint degrees of freedom {u j }and {u

j +1 / 2 }, for all j Z , respectively:

10 + 2(1 ) u j 1 + u j +1 + 140 22(4 + ) u j + 22( 1) 80 (u j 1/ 2 + u j +1 / 2 ) = 0(2.17)

and

2 ( 1) 40 [u j 1 + u j +1 ] + 80 22( + 4) u j +1 / 2 = 0with = h. We use the latter relation to express the midpoint degree of freedomu j +1 / 2 in terms of the nodal degrees of freedoms u j 1 and u j +1 as follows:

u j +1 / 2 =2( 1) 40

2(2( + 4) 40)u j + u

j +1 , j Z .


7/26


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11.5

1

0.5

0

0.5

1

1.5

2

x

U

Numerical real wave obtained using the spectral element schemeNumerical real wave obtained using the finite element schemeNumerical real wave obtained using the optimal schemeExact real wave

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.51.5

1

0.5

0

0.5

1

1.5

2

x

U


Fig. 2.1 . Variation of the numerical approximations of the solution with linear spectral, nite,and optimal schemes to (2.16) using h = 1 along the lines (a) y = x and (b) y = 2 x.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11.5

1

0.5

0

0.5

1

1.5

2

x

U


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.51.5

1

0.5

0

0.5

1

1.5

2

x

U


Fig. 2.2 . Variation of the numerical approximations of the solution with linear spectral, nite,and optimal schemes to (2.16) using h = 0 .67 along the lines (a) y = x and (b) y = 2 x.

Now substituting these values into (2.17), we obtain a single equation corresponding

to nodal degrees of freedom:4( 1) + 2 2(3 8) 240

2(2( + 4) 40)u j 1 + u

j +1

+4(2 + 3) 22(3 + 52) + 240

2( + 4) 40u j = 0 , j Z .

This relation is of the same form as (2.3) and (2.5), and we may therefore proceedas before by inserting a nontrivial solution of the form u j = eij

(2) h into the above

equation. Proceeding as before, we arrive at the expression

(2.18) cos( (2) h) =4 (2 + 3) 22(3 + 52) + 240

4(1 ) 22(3 8) + 240,

and hence, for 1, there exists a solution of the form

(2.19) (2) h = +3 22880

5 +63 2 126 + 88

24192007 + .


8/26


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11.5

1

0.5

0

0.5

1

1.5

2

x

U


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.51.5

1

0.5

0

0.5

1

1.5

2

x

U


(a) y = x (b) y = 2 x

Fig. 2.3 . Variation of the numerical approximations of the solution with quadratic spectral, nite, and optimal schemes to (2.16) using h = 2 .5 along the lines (a) y = x and (b) y = 2 x.

For = 0 and = 1, the above expression reduces to the ones obtained for niteelement [1] and spectral element [2] schemes. The choice = 2 / 3 means the rst termof the above expression vanishes and gives two additional orders of accuracy in thephase compared with the standard schemes. Furthermore, the absolute value of thecoefficient of the leading term with the optimum value of is decreased by factors of 50 and 25 compared to the leading coefficient obtained with the nite and spectral

element schemes, respectively.We can extend the scheme to higher numbers of spatial dimensions in preciselythe same way as we described earlier for linear elements, provided that a suitablenonstandard quadrature rule can be identied. One obtains the optimally blendedscheme in the case of quadratic elements if the following quadrature rule,

(2.20)

1 1 f (x)dx 13(3 + 2 ) 5f 15(3 + 2 ) +4(2+3 )f (0)+5 f 15 (3 + 2 ) ,is used to approximate the entries in the mass and stiffness matrices. Moreover, for theoptimum value of = 2 / 3, (2.20) reduces to the quadrature rule given in [3]. In Figure2.3, we show the effect of using piecewise quadratic elements instead of piecewiselinear elements along the lines y = x and y = 2 x relative to the bottom edge 1 .As expected, the numerical approximations corresponding to the nite and spectralelement schemes are, respectively, leading and lagging even with quadratic elements,but again the optimal scheme performs much better even when h is relatively large.The results obtained by reducing the size of the elements are given in Figure 2.4.

2.6. Extension to cubic elements. Turning to the case of cubic elements, wehave the following expression for the discrete wave number for the blended scheme:

(3) h = +4 3604800

7 +4 2 15 + 11

635040009 + ,

where the rst term vanishes corresponding to the optimum value of the blendingparameter = 3 / 4, and we again observe that two additional orders of accuracy areachieved.


9/26


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11.5

1

0.5

0

0.5

1

1.5

2

x

U


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.51.5

1

0.5

0

0.5

1

1.5

2

x

U

Numerical real wave obtained using the spectral el ement schemeNumerical real wave obtained using the finite element schemeNumerical real wave obtained using the optimal schemeExact real wave

(a) y = x (b) y = 2 x

Fig. 2.4 . Variation of the numerical approximations of the solution with quadratic spectral, nite, and optimal schemes to (2.16) using h = 2 along the lines (a) y = x and (b) y = 2 x.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11.5

1

0.5

0

0.5

1

1.5

2

x

U


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.51.5

1

0.5

0

0.5

1

1.5

2

x

U


(a) y = x (b) y = 2 x

Fig. 2.5 . Variation of the numerical approximations of the solution with cubic spectral, nite,and optimal schemes to (2.16) using h = 5 along the lines (a) y = x and (b) y = 2 x.

Once again, we can extend the scheme to higher numbers of spatial dimensionsprovided a suitable quadrature rule is available. For cubic (and higher order) elements,no such rule seems to be known in the literature. However, we may use the followingnew quadrature rule (which is a special case of Theorem 2.1):

(2.21) 1 1 f (x)dx 7840681 (f (+ ) + f (+ ))(39 + 681)(681 3) + (f ( ) + f ( ))(39 681)(3 + 681) ,where =

2730 70681/ 70, to approximate entries in the stiffness and mass

matrices which gives us the optimally blended scheme in the case of cubic elements.The numerical approximations obtained with piecewise cubic elements are shown inFigures 2.5 and 2.6 for four and ve cubic elements in each direction, and once againthe optimal scheme performs noticeably better compared to the nite and spectralelement schemes.

2.7. Extension to arbitrary order elements. The question naturally arisesof how the above results extend to elements of arbitrary order. In Theorem 3.2 we


10/26


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11.5

1

0.5

0

0.5

1

1.5

2

x

U


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.51.5

1

0.5

0

0.5

1

1.5

2

x

U


(a) y = x (b) y = 2 x

Fig. 2.6 . Variation of the numerical approximations of the solution with cubic spectral, nite,and optimal schemes to (2.16) using h = 4 along the lines (a) y = x and (b) y = 2 x.

show that for elements of order p N , the discrete wave number for the blendedscheme is given by

( p) h = + 1 +1 p 1 F

( p) () + C ( p) F ( p+1) () + O()2 p+5 ,

where

F ( p) () and C ( p) are dened in Theorem 3.2. In the case = 0 , this result

agrees with Theorem 3 .2 in [1], while in the case = 1 we obtain the result given inTheorem 4 .2 of [2]. One immediate consequence of this new result is that the optimalblending parameter is given by = p p+1 . With this choice, we obtain

( p) h = + C ( p) F ( p+1) () + O()2 p+5 ,

showing that in general we obtain two additional orders of accuracy with the optimalchoice of blending parameter . Moreover, in Corollary 3.3, we show that the absolutevalue of the leading coefficient in the error ( p) h is considerably reduced by theuse of blending. The proof of these statements forms the topic of section 4.

2.8. Nonstandard quadrature rule for elements of arbitrary order. Theuse of such nonstandard quadrature rules in the implementation of the optimallyblended scheme is rather attractive in practice and provides a simple way to extendthe blended schemes to a higher number of spatial dimensions. More specically itmeans that an existing, standard nite element code can be adapted to implement theoptimally blended scheme merely by replacing the usual Gaussian quadrature rule by the nonstandard quadrature rule . Unfortunately, the existence of suitable nonstandardquadrature rules for general pth order elements does not seem to be available in theexisting literature.

If we denote the bilinear form for the nite and spectral elements schemes byB (, ) and B (, ), respectively, then the bilinear form for the blended scheme is givenby(2.22) B (u, v ) = (1 )B (u, v ) + B (u, v )for piecewise polynomials u and v. The difference between the bilinear forms for thenite and spectral element schemes lies in the fact that the spectral element scheme


11/26


uses the GaussLobatto quadrature rule which we denote by Q( p) to evaluate theintegrals, while the nite element scheme evaluates (via GaussLegendre quadrature)the integrals exactly. Consequently, the bilinear form for the blended scheme (2.22)

should be based on a quadrature rule Q( p)

for which

Q( p) (f ) = (1 ) 1 1 f (x)dx + Q( p) (f ) f P 2 p+1 ,where P 2 p+1 denotes the space of polynomials of degree 2 p + 1. The following resultconstitutes the extension of the nonstandard quadrature rules to elements of arbitraryorder.

Theorem 2.1. Let [0, 1) be xed, and denote the zeros of L p+1 L p 1 by {i}

pi=0 . Then {i}

pi=0 are distinct and contained in (1, 1). Let {wi}

pi=0 denote the

weights dened by the rule

(2.23) wi =2[ p(1 + ) + ]

p( p + 1) L p(i )[L p+1 (i ) L p 1(i )]i = 0 , 1, . . . , p .

Then, the weights {wi} pi=0 are well-dened and positive. Moreover, if we use the

weights and nodes to dene a ( p + 1) -point quadrature rule Q( p) , then Q

( p) satises

the following identity:

(2.24)

Q( p) (f ) = (1

)

1

1

f (x)dx +

Q( p) (f ) f P 2 p+1 ,

where Q( p) is the ( p+ 1) -point GaussLegendreLobatto quadrature rule dened in [2,eq. (2.7)] . Consequently, Q

( p) is exact for all f P 2 p 1 .

The fact that the nodes are distinct means that the term L p+1 (i ) L p 1(i )appearing in the denominator for the quadrature weights (2.23) is nonzero. Moreover,the factor L p(i ) is also nonzero. This can be seen by rst noting that, thanks to therecurrence relation for Legendre polynomials,

2 p + 1

p + 1

i L p(i ) = L p+1 (i ) +p

p + 1

L p 1(i )

= +p

p + 1L p 1(i ).

If i is nonzero, then the right-hand side is nonzero (otherwise we have L p 1(i ) =L p(i ) = L p+1 (i ) = 0 which would imply all subsequent Legendre polynomials havea common zero), and hence L p(i ) is nonzero. Equally well, if i does vanish, thenL p 1(i ) also vanishes meaning that p is even, and hence L p(0) is nonzero.

The proof of this result is given in section 4. Observe that for = 0 and = 1,

Q( p) reduces to the standard GaussLegendre and GaussLegendreLobatto rules,

respectively.In Table 2.1, nodes and corresponding weights of the optimal quadrature rule

Q( p) are given for the optimum value of the blending parameter = p/ ( p + 1). It

is a simple matter to compute the higher order rules using the expressions given inTheorem 2.1.

The quadrature rule Q( p) can then be used to extend the one dimensional scheme

to higher dimensions for elements of arbitrary order as described in section 2 .3 for


12/26


Table 2.1Nodes and corresponding weights of the optimal quadrature rule Q ( p ) for = p/ ( p + 1) and

orders p = 1 , . . . , 4.

Order p Abscissas i Weights wi

1 0.8164965809 1

2 0 1.2307692308 0.9309493363 0 .3846153846

3 0.9643352759 0 .1998260144 0.4293520583 0 .8001739855

4 0 0.6937669377 0.9783156780 0 .1217872771 0.6387313983 0 .5313292541

rst order elements. Using the same arguments used there leads to the conclusionthat the wave number of the discrete solution is given by khp , where

k2hp = ( p) (kx )

2 + ( p) (ky )2 + ( p) (kz )

2 ,

where k2x + k2y + k2z = 2 . Thanks to (3.2) given in Theorem 3 .1, we obtain

k2hp

= 2 +p!

(2 p)!

2

1 +1

p 1

h2 p

2 p + 1[k2 p+2

x+ k2 p+2

y+ k2 p+2

z]

+ O(h2 p+2 2 p+4 ),which is valid for general and for all p 2. For the optimal choice of = p/ ( p+1),using Corollary 3.3, we havek2hp =

2 +8

(2 p1)( p + 1)!(2 p + 2)!

2 h2 p+2

2 p + 3[k2 p+4x + k

2 p+4y + k

2 p+4z ] + O(h2 p+4 2 p+6 ).

3. Analysis of dispersion for elements of arbitrary order. Our rst resultgives the discrete dispersion relation for the blending of spectral-nite element ap-proximation for elements of arbitrary order p N , with blending parameter [0, 1],and generalizes the particular cases given in section 2. The proof of the followingtheorem follows exactly the same arguments used in [2] for the proof of Theorem 4.1and is therefore omitted.

Theorem 3.1. Let > 0 and consider the sequences {a p} p=1 and {b p} p=1 dened by the recursion relations

(3.1)a p+1 =

2 p + 1

b p + a p 1

b p+1 = 2 p + 1 a p + b p 1

for p N with a0 = 1 , a1 = 1 , b0 = 0 , and b1 = 1 /. Then, the discrete dispersion relation for the optimal scheme of order p N is given by

(3.2) cos ( p) h = R( p) (2) = ( 1) p

( p)1 () + ( p)2 ()

( p)1 () ( p)2 ()

,


13/26


where = h , and

( p)1 () = a p (b p 1( ( p + 1) + p) + p(2 p + 1) a p)

and

( p)2 () = b p (a p 1 ( ( p + 1) + p) p(2 p + 1) b p) .The sequences {a p} p=1 and {b p} p=1 originally appeared in Lemma 5 .2 of [2] in theanalysis of the pure spectral element scheme. In particular, we note that the denom-inator appearing in (3.2) is nonvanishing. This can be seen by rst noting that

( p)1 () ( p)2 () = p(2 p + 1)( a

2 p + b2 p),

for 1, and then using the following identity,

(3.3) sin( n/ 2) cos( n/ 2)cos( n/ 2) sin( n/ 2)anbn

= 2 J n +1 / 2()Y n +1 / 2() ,proved in [2] to see that

a2 p + b2 p =

2

J p+1 / 2 ()2 + Y p+1 / 2()2 ,

where J and Y are cylindrical Bessel functions of the rst and second kinds, respec-tively [5]. Finally make use of the identity in the rst equation of (8 .479) of [5] todeduce that

a2 p + b2 p > 1

for all and p N .In the case when = 1, the expression (3.2) reduces to the discrete dispersion

relation (4 .6) obtained in [2] for spectral element methods, while in the case = 0expression (3.2) gives an alternative form of the discrete dispersion relation (3 .2)obtained in [1] for nite element methods. As pointed out in [1], R ( p) (2) is a rationalfunction of degree [2 p/ 2 p] in for all p N which, in the case of the pure nite elementmethod ( = 0) , corresponds to certain types of Pade approximants.

The following theorem proved in section 4 gives the leading term for the error inthe discrete dispersion relation for the blended scheme with parameter [0, 1].

Theorem 3.2. Let p 2 and [0, 1]. Then, the error in the discrete dispersion relation (3.2) is given by

cos ( p) h cos = 1 1 +1 p F

( p) () C ( p) F ( p+1) () + O()2 p+6

or, if is sufficiently small,

(3.4) ( p) h = 1 +1 p 1 F

( p) () + C ( p) F ( p+1) () + O()2 p+5 ,where

C ( p) = 2 (2 p + 3)

(2 p1)1 +

1 p

2

(2 p + 3) 1 +1 p

+ 22 p2 + p + 1

2 p1


14/26


and

F ( p) (s) =12

p!(2 p)!

2 s2 p+1

2 p + 1.

As expected, when = 0 or = 1, the above result reduces to spectral and niteelement schemes, respectively. More interestingly, (3.4) indicates that the blendingterm in the error can be eliminated by choosing = p/ ( p+1), resulting in an additionaltwo orders of accuracy in the discrete dispersion relation.

Corollary 3.3. Let p 2. For = p/ ( p+1) , the error in the discrete dispersion relation (3.2) is given by

(3.5) ( p)

h

=

4

(2 p1)( p + 1)!

(2 p + 2)!

2 2 p+3

2 p + 3+

O()2 p+5 .

Proof . Substitute = p/ ( p + 1) in (3.4), and applying straightforward manipula-tions, we obtain (3.5) as required.

In Table 3.1 we give closed form expressions for the rational function R ( p) ()obtained from Theorem 3.1 along with the leading terms in the error for 1obtained from Theorem 3.2 for orders p from 1 up to 3 and [0, 1]. Moreover, theerror in the leading term for the optimum value of , i.e., = p/ ( p+1) obtained fromCorollary 3.3 is given for a small wave number limit.

Table 3.1The discrete dispersion relation R ( p ) () = cos ( p ) h for order p approximation given in The-

orem 3.1. We also indicate the leading term in the series expansion for the error when 1 for both general [0, 1] (see Theorem 3.2) and = p/ ( p + 1) (see Corollary 3.3) for p 2.

Order p R ( p ) ()

12 ( + 2) 62 ( 1) 6

24 (2 + 3) 22 (3 + 52) + 240

4 (1 ) 22 (3 8) + 240

36 (3 + 4) 44 (26 + 135) + 240 2 ( + 48) 25200

6 ( 1) + 2 4 (8 15) + 120 2 (2 9) 25200

Order p cos 1 R ( p ) () cos 1 R( p ) () , = p/ ( p + 1)

13 (2 1)

24+ O (5 )

5

480

25 (3 2)

2880+ O (7 )

7

75600

37 (4 3)

604800+ O (9 )

9

31752000

We make the following observations regarding the optimally blended scheme:1. The leading term is two orders more accurate compared with the standard

spectral and nite element schemes; see [16, 1, 2, 8], where the leading term in the


15/26


102

101

100

101

1015

1010

10 5

100

105

| ( R

( p ) (

) )

2

( c o s (

) )

2 | , = 0 , 1 , p

/ ( p +

1 )

Finite element schemeSpectral element schemeOptimal scheme

4

1

1

6

10 2 10 1 10 0 10 110 15

10 10

10 5

10 0

10 5

| ( R

( p ) (

) )

2

( c o s (

) )

2 | , =

0 , 1 , p

/ ( p +

1 )


6

1

1

8

(a) p = 1 (b) p = 2

10 2 10 1 10 0 10 110

15

1010

105

10 0

105

| ( R

( p ) (

) )

2

( c o s (

) )

2 | , =

0 , 1 , p

/ ( p +

1 )

Finite element schemeSpectral element schemeOptimal Scheme

1

1

10

8

10 2 10 1 10 0 10 110

15

1010

105

100

105

| ( R

( p ) (

) )

2

( c o s (

) )

2 | , =

0 , 1 , p

/ ( p +

1 )


1

12

1

10

(c) p = 3 (d) p = 4

Fig. 3.1 . Error in discrete dispersion relations of orders p = 1 to 4 versus wave number for nite, spectral, and optimal schemes. For pth order nite and spectral element schemes the slope of the lines is 2 p + 2 , whereas for optimal scheme the slope of the line is 2 p + 4 .

expressions was accurate to order O()2 p . This is illustrated in Figure 3.1, whereit is observed that for a pth order scheme the slope of the lines with spectral andnite element schemes is 2 p + 2, whereas with the optimal scheme the slope is2 p + 4 .

2. The coefficient of the leading term in the error obtained with the blendedscheme for the optimum value of is 2/ (4 p2 1)(2 p + 3) and 2 p/ (4 p2 1)(2 p + 3)times better compared with the leading terms in the error obtained with nite [1] andspectral [2] element schemes, respectively. This is also illustrated in Figure 3.1, whereit is observed that the absolute value of the error for the optimized scheme is superiorto that of the standard schemes even for modest values of .

3. Figure 3.2 shows the frequency spectra of the nite element, spectral element,and optimally blended scheme for elements of order p = 1 , . . . , 4. In particular, onecan see the so-called cutoff frequencies of the schemes corresponding to the values of at which the magnitude of the rational function R ( p) () becomes greater than unity.For such frequencies, the discrete wave number ( p) is imaginary and the discretewaves cease to propagate. These frequencies can be computed explicitly for the rstorder elements p = 1 by considering when the inequality |R

( p) ()| > 1 is satised.

Inserting the expression for R ( p) () from Table 3.1 reveals that the inequality holds


16/26


0 0.5 1 1.5 2 2.5 3 3.5 40

0.5

1

1.5

2

2.5

3

3.5

4

Normalized Wavenumber real( (1)h)

N o r m a

l i z e

d F r e q u e n c e y

Finite element schemeSpectral element schemeOptimal schemeExact continuum

= 3.46

= 2.00

= 2.45Cutoff frequencies

0 1 2 3 4 5 6 70

1

2

3

4

5

6

7

8

9

10


N o r m a

l i z e



(a) p=1 (b) p=2

0 1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

14


N o r m a

l i z e



0 2 4 6 8 10 12 140

2

4

6

8

10

12

14

16

18

20


N o r m a

l i z e



(c) p=3 (d) p=4

Fig. 3.2 . Frequency spectra of the one dimensional spectral, nite, and optimally blended scheme for elements of order p = 1 , . . . , 4.

for greater than

(3.6) = 121 + 2 .For the nite element scheme ( = 0) and spectral element scheme ( = 1) we obtaincutoff frequencies of 23 and 2, respectively, in agreement with the results presentedin [16]. For the optimally blended scheme, the cutoff frequency is given by 6. Thesefrequencies are indicated in Figure 3.2(a). All of the schemes corresponding to p = 1elements fail to admit propagating waves for higher frequencies and thus have a singlestopping band extending to innity. In general, the schemes based on elements of

order p have p stopping bands, as shown in Figure 3.2(b)(d).The so-called spatial resolution limit is dened [16] to be the number of elementsper wavelength corresponding to the cutoff frequency. The spatial resolution limit for p = 1 elements is obtained by inserting (3.6) into (2.8) to obtain

cos (1) h = R(1) 121 + 2 = 1


17/26


or

(1) h = ,

giving a spatial resolution limit for the p = 1 elements of

h

=2

(1) h=

2

= 2 .

Examination of Figure 3.2(b)(d) reveals that for a pth order scheme, the spatial reso-lution limit is given by 2 /p elements per wavelength in agreement with the observationof [16].

4. Proofs of the results. This section provides the proofs of general results forthe error in the discrete dispersion relation for the blended scheme.

4.1. Basic polynomials. Let p N be given and [0, 1] be a parameter tobe determined. Dene the bilinear form

(4.1) B (v, w) = (1 ) 1 1(v w 2vw)dx + Q( p) (v w 2vw),where Q( p) is the ( p+1) point GaussLobatto quadrature rule and > 0 is a constant.If v, w P p , then v w P 2 p 2 and the quadrature rule Q( p) integrates this productexactly. Hence, if v, w P p, then

(4.2) B (v, w) = 1

1v w dx 2 (1 )

1

1vwdx + Q( p) (vw) .

We introduce basic polynomials p, p P p for p N dened in [2] by

(4.3) p(1) = 1 , p(1) = ( 1) p+1 : B ( p , w) = 0 w P p H 10(1, 1)and

(4.4) p(1) = 1 , p(

1) = (

1) p : B ( p, w) = 0 w P p

H 10(

1, 1).

Observe that if is sufficiently small, then (4.3) and (4.4) admit a unique solution.From (4.3) and considering the parity of p, we deduce that p P p 1 for all p N .Moreover, w p and w p P 2 p 1 , for w P p H 10(1, 1), and it follows that thequadrature rule Q( p) is exact in (4.1) for w = p , so that

B ( p, w) = 1 1 pw 2 pw dx = 0 w P p H 10(1, 1).Hence, for p the bilinear form (4.1) coincides with the bilinear form

B (v, w) = Q( p) (v w ) 2Q( p) (vw)considered in [2], and we may, therefore, for p quote results for p directly from [2].In particular, from Theorem 5 .1 of [2], we have

B ( p, p) = B ( p , p) = 2a pb p

,


18/26


where a p and b p are dened in Theorem 3.1. We shall require the correspondingexpression for B ( p, p).

Theorem 4.1. Let p = 2 , 3, 4, . . . . Then

(4.5) B ( p, p) = 2 pa p+1 + ( p + 1) a p 1 pb p+1 + ( p + 1) b p 1

,

where {a p}and {b p}are dened in Theorem 3.1 and 2 = .Proof . For the duration of this proof the superscript on p will be omitted since noconfusion is likely to arise. Suppose w P p 1 H 10(1, 1), then w P 2 p 1 . Usingthe fact that the quadrature rule in the bilinear form (4.1) is exact for functionsbelonging to P 2 p 1 , and using denition (4.4), we obtain

(4.6) 1

1( + 2)wdx = 0 w P p 1 H

10(1, 1).

Now, we can write F (x) = (x) + 2(x) P p in the form

(4.7) (x) + 2(x) = p+1

j =1

j L j (x),

where j are the scalars and L j is the Legendre polynomial of degree j . Now insertingw(x) = (1 x2 )L (x) for 1 p 2 together with (4.7) into (4.6), we obtain 1 = 2 = = p 2 = 0 . Also, parity considerations imply that p = 0. Hence,(4.8) F (x) = (x) + 2(x) = p+1 L p+1 (x) + p 1L p 1(x).

Now, choosing w(x) = (1 x2 )L p 1(x) P p H 10 (1, 1) in denition (4.4), we get

(4.9) 1 1 F (x)w(x)dx 2 1 1 (x)w(x)dx Q( p) (w) = 0 .Also, using (4.8) together with the rst term of the last expression gives

1 1 F (x)w(x)dx = 1 1(1 x2) p+1 L p+1 (x)L p 1 (x) + p 1[L p 1(x)]2 dx,and then exploiting the orthogonality property of the Legendre polynomials, we obtain

(4.10) 1 1 F (x)w(x)dx = 2 p( p1)2 p1 p 1 .The error in the GaussLobatto quadrature rule is given by

(4.11) E = 1

1(x)w(x)dx Q( p) (w) =

( p + 1) p322 p+1 [( p 1)!]4(2 p + 1)[(2 p)!]3

D 2 p(w),

using (4 .10-27) from [14]. Now using the Leibniz rule, we get

(4.12) D 2 p(w) =(2 p)! p!2

( p) (0)w( p) (0) ,


19/26


where

(4.13) ( p) (0) = p+1

2L ( p+1) p+1 (0)

obtained by differentiating (4.8) p times with respect to x. Moreover,

w( p) (x) = D p (1 x2 )L p 1(x) .Now using the identity D p (1 x2 )L p 1(x) = p( p1)L

( p 1) p 1 (x), we get

(4.14) w( p) (0) = p( p1)L( p 1) p 1 (0) .

Hence, using L ( p) p (0) =(2 p)!

p!

1

2 p, together with (4.13) and (4.14), (4.12) simplies to

give

D 2 p(w) = p+1

2(2 p + 1)( p1)[(2 p)!]3

p322 p(2 p1)[( p1)!]4.

Now substituting this value into (4.11) and performing ordinary manipulations gives

(4.15) E = 1 1 (x)w(x)dx Q( p) (w) = 22 ( p2 1)2 p1 p+1 .Inserting the values from (4.10) and (4.15) into (4.9), we obtain p 1 =

( p+1) p p+1 .

Consequently, (4.8) can be rewritten in the form

(4.16) F (x) = (x) + 2(x) = p+1 1 +1 p

L p 1(x) + L p+1 (x) .

Observe that we may write

(4.17) ( x) = p(x) + p 2(x)

for suitable constants and where is given in [2, 5.10] and dened as

(4.18) p(x) = p/ 2

j =0

12

j +1

L(2 j +1) p+1 (x)

and satises the property

(4.19) p (x) + 2 p(x) = L p+1 (x).

Consequently, we have

F (x) = p (x) + 2 p(x) + p 2(x) +

2 p 2(x) ;

using the property (4.19), we get

F (x) = L p+1 (x) L p 1(x).


20/26


Comparing the last equation with (4.16), we are led to the choices = p+1 and = p+1 ( p + 1) /p , and with these values, (4.17) becomes

(4.20) ( x) = p+1 1 +1 p p 2(x) + p(x) .

Applying the boundary condition (1) = 1, we obtain p+1 = p/ (1), providedthat (1) is nonzero, with (1) = ( p +1) p 2(1)+ p p(1) . Consequently, may bewritten in the form

(4.21) ( x) =(x)(1)

.

We want to obtain a closed form expression for (4.1), and for this we dene I (x) =

(x + 1) / 2 + ( 1) p

(1 x)/ 2, soB (, ) = B (, I ) + B (, I )

= [ I ]1 1 1 1( + 2) I dx,and applying integration by parts together with (4.16), we get

B (, ) = 2 (1) 2 p+1 1 + 1 +1 p

= 2(1)

[ (1) + ( p + ( p + 1))] .(4.22)

Now, as in [1, 2], using the values of and its derivatives at the boundary x = 1in terms of the series a p+1 , a p 1 , b p+1 , and b p 1 which can be obtained from therecurrence relation (3.1) and are proved in [2], (1) and (1) can be written as

(1) = 1

( ( p + 1) b p 1 + pb p+1 )

and (1) + ( p + 1) + p = ( p + 1) a p 1 + pa p+1 .

Hence, inserting the above values into (4.22) and simplifying gives

(4.23) B ( p, p) = 2 ( p + 1) a p 1 + pa p+1 ( p + 1) b p 1 + pb p+1

which completes the proof.For p we rewrite expressions (5 .20) and (5 .22) given in [2] in terms of

F ( p) (2)

with higher order terms as the quadrature rule is exact for p in the bilinear form(4.1). Moreover, the estimates (5 .20) and (5 .22) are the same as (4 .16) and (4 .15)in [1], for p = 2 N and p = 2 N + 1, respectively. Therefore, when p is even and = ( m + 1 / 2), m Z , then

E ( p) (2) = 2 F

( p) (2) 2

(2 p + 1) 2

2 p1F ( p+1) (2)

+ O(22 p+4 ),


21/26


and when p is odd and = m,m Z , then

E ( p) (2) = 2 F ( p) (2)

(2 p + 1) 2

2 p

1

2F ( p+1) (2) + O(22 p+6 ).Theorem 4.2. Let p N satisfy p 2. Then 1. if p is even, and = ( m + 1 / 2), m Z , then

(4.24) E ( p) (2) = 2 1 +

1 p F

( p) (2) 2F ( p+1) (2)C ( p) + O(22 p+6 );2. if p is odd, and = m,m Z , then

E ( p) (2) =

2 1 +

1

pF ( p) (2)

2F ( p+1) (2)

C ( p) +

O(22 p+4 ),(4.25)

where

C ( p) = 2 1 +

1 p

2 2 p + 32 p 1

1 +1 p

(2 p + 3) 1and

F ( p) (2) =12

p!(2 p)!

2 (2)2 p+1

2 p + 1p 2.

Proof . First, consider the case when p is even. Writing the series {a p}and {b p}in (4.23) in terms of Bessel functions using the identity (3.3), we getB ( p, p) =

2 ( p + 1)( J p 1/ 2 ()cot + Y p 1/ 2()) p(J p+3 / 2()cot + Y p+3 / 2()) ( p + 1)( J p 1/ 2 () Y p 1/ 2()cot ) p(J p+3 / 2() Y p+3 / 2()cot )

,(4.26)

where J and Y are cylindrical Bessel functions of the rst and second kinds, respec-tively, and = ( m + 1 / 2) for all m Z . Adding 2 tan to (4.26) and applying

straightforward manipulations give

(4.27) B ( p, p) + 2 tan = 2

cos2 Q p+3 / 2 () 1 Q p+3 / 2 ()tan

1,

where

(4.28) Q p+3 / 2 () = ( p + 1) J p 1/ 2 () pJ p+3 / 2() ( p + 1) Y p 1/ 2 () pY p+3 / 2()

,

and for small , i.e., when 1 is given by

(4.29) Q p+3 / 2 () = 1 +1 p F

( p) (2) + C ( p) F ( p+1) (2) + .With the aid of this estimate, we obtain that

E ( p) (2) = 2 1 +

1 p F

( p) (2) 2F ( p+1) (2)C ( p) + .


22/26


The assertions concerning polynomials of odd order are proved in a similar fashion.Again, using the identity (3.3), we write the series {a p}and {b p}in (4.23) in termsof Bessel functions and get

B ( p , p) =

2 ( p + 1)( Y p 1/ 2 ()cot J p 1/ 2()) p(Y p+3 / 2 ()cot J p+3 / 2 ()) ( p + 1)( J p 1/ 2() + Y p 1/ 2 ()cot ) p(J p+3 / 2 () + Y p+3 / 2 ()cot )

,(4.30)

where = m for all m Z . Subtracting 2 cot from (4.30) and after simplication,we obtain

(4.31) B ( p, p) 2 cot = 2

sin2 Q p+3 / 2 () 1 + Q

p+3 / 2 ()cot

1.

Now, using (4.29) in the above expression and simplifying gives

E ( p) (2) = 2 1 +

1 p

F ( p) (2) 2F

( p+1) (2)

C ( p) + as required.

4.2. Proof of Theorem 3.2. We now prove Theorem 3.2 by using expressionsderived in [2] for cos ( p) h cos2 valid for small wave numbers; i.e., 1 for botheven and odd order polynomials.

Proof . First consider the case when p is even. We reconsider expression

(4.32) cos ( p) h cos2 = 2E ( p) + E

( p) +

obtained in [2] for small ; i.e., 1. Now,

2E ( p) = 2 F ( p) (2) 2F ( p+1) (2)

(2 p + 1) 2

2 p1+

and

E ( p) (2) = 2 1 + 1 p F

( p) (2) 2F ( p+1) (2)C ( p) + .

Inserting these values into (4.32) and simplifying gives

(4.33) cos ( p) h cos2 = 2 1 1 +1 p F

( p) (2) 2C ( p) F ( p+1) (2) + ,

where C ( p) = 2 (2 p+3)(2 p 1) (1 +1 p )

2 (2 p + 3)(1 + 1 p ) + 2 2 p2 + p+12 p 1 .

For the case when p is odd, we reconsider

(4.34) cos ( p) h cos2 = 2E ( p) + E

( p) +

obtained in [2], where

2E ( p) = 2 1 +

1 p F

( p) (2) 2F ( p+1) (2)C ( p) +


23/26


and

E ( p) (2) = 2 F ( p) (2) 2F ( p+1) (2)

(2 p + 1) 2

2 p

1

+ .Now, substituting these values into (4.34) and simplifying results in (4.33), which iswhat was required. Finally, for small ( p) h, the approximation

cos ( p) h cos2 = 2(( p) h 2) + gives us the required estimate (3.4).

4.3. Proof of Theorem 2.1.Proof . Let f P p be written in the form

f (x) = p

j =0j (x)f (j ),

where j P p satises j (k ) = jk j = 0 , 1, 2, . . . , p . Applying the quadrature rule

Q( p) gives

Q( p) (f ) = p

j =0

wj f (j ),

where {j } pj =0 and {wj }

pj =0 are the nodes and weights of Q

( p) , respectively. Later,

we show that the quadrature weights dened in (2.23) satisfy

(4.35) wj = 1 1 j (x)dx.Hence, for f P p

Q( p) (f ) =

p

j =0

f (j )

1

1j (x)dx =

1

1

p

j =0

f (j ) j (x)dx =

1

1f (x)dx,

and so Q( p) is exact for all f P p. Now, let f P 2 p 1 be written in the form

f (x) = pf (x) + (x)q(x)

for q P p 2 , where (x) = L p+1 (x)L p 1(x) and pf P p denotes the interpolentto f at the nodes {j } pj =0 . Integrating the above equation and using the orthogonality

property of the Legendre polynomials, we get

1

1f (x)dx =

1

1 pf (x)dx.

Moreover, since vanishes at the nodes of Q( p) , we can write

1 1 pf (x)dx = Q( p) ( pf ) = Q( p) ( pf + q ) = Q( p) (f ),


24/26


and it follows that Q( p) is exact for all f P 2 p 1 . Since the GaussLegendreLobatto

rule is also exact for f P 2 p 1 , it follows that

(4.36) Q( p) (f ) = (1 )

1

1 f (x)dx + Q( p)

(f ) f P 2 p 1 ,

and hence identity (2.24) holds for f P 2 p 1 .Now let f P 2 p be written in the form

f (x) = (x)L p 1 (x) + q(x)

for suitable constant R and q P 2 p 1 . Since vanishes at the quadrature pointsof Q

( p) , we have

(4.37) Q( p) (f ) = Q( p) (q) = (1 ) 1

1q(x)dx + Q( p) (q),

where the second step follows from (2.24) applied to q P 2 p 1 . The orthogonalityproperty of Legendre polynomials means that

(1 ) 1 1 (x)L p 1 (x)dx = (1 ) 1 1(L p+1 (x) L p 1 (x))L p 1 (x)dx= (1 )

1

1

L2 p 1(x)dx.

Furthermore, since L p+1 and L p 1 coincide at the nodes of the GaussLobatto quadra-ture rule, we have

Q( p) ( L p 1) = Q( p) ([L p+1 L p 1]L p 1) = (1 )Q( p) (L2 p 1),and since the GaussLobatto rule has precision 2 p1, we obtain

Q( p) ( L p 1) = ( 1) 1 1 L2 p 1(x)dx.Consequently, we deduce that

(1 ) 1 1 (x)L p 1(x)dx + Q( p) ( L p 1) = 0 ,and then adding times this identity to (4.37) shows that

Q( p) (f ) = (1 ) 1 1 f (x)dx + Q( p) (f )for all f P 2 p. It is trivial to see that (2.24) now holds for all f P 2 p+1 since bothsides of (2.24) vanish identically when f is the odd function f (x) = x2 p+1 .

The positivity of the weights can be seen by inserting f (x) = 2j (x) P 2 p into(2.24) to obtain for [0, 1)

wj = (1 ) 1 1 2j (x)dx + Q( p) ( 2j ) > 0.


25/26


We now show that the nodes {i} pi=0 are real, are distinct, and lie within the

interval ( 1, 1). Suppose this were not the case. Let {i}mi=0 with m < p be thepoints where (x) P p+1 changes sign in ( 1, 1), and then the polynomial W (x) =(x 0)(x 1) (x m ) (x) vanishes at the nodes of Q

( p)

but does not changesign in (1, 1); i.e.,

(4.38) 1 1(x 0)(x 1 ) (x m ) (x)dx = 0 .Hence, thanks to (2.24) applied to W P 2 p+1 , we obtain

0 = Q( p) (W ) = (1 ) 1 1 W (x)dx + Q( p) (W ),but the right-hand side is nonzero since W does not change sign, and we obtain acontradiction. Hence m = p.

Above, we have shown that (2.24) holds provided that the weights satisfy (4.35).We now show that choosing the weights according to (4.35) implies (2.23) holds.Observe that J (x) = (x)/ (x J ) (J ) P p so that

(4.39) wJ = 1 1 J (x)dx = 1(J ) 1 1 (x)x J dx.We recall the ChristoffelDarboux identity [14, 4.7-3]:

p

k =0

Lk (x)Lk (y)(2k + 1) =L p+1 (x)L p(y) L p(x)L p+1 (y)

x y( p + 1) , x = y.

Choose y = J and integrate from 1 to 1 with respect to x to get

(4.40) 1 1 L p+1 (x)L p(J ) L p(x)L p+1 (J )x J dx = 2 p + 1 , p N .Now, inserting L p+1 (x) = (x) + L p 1 (x) and L p+1 (J ) = L p 1 (J ) gives

2 p + 1

= L p(J ) 1 1 (x)x J dx + 1

1

L p 1(x)L p(J ) L p(x)L p 1(J )x J

dx,

and then using (4.40), we obtain

2 p + 1

= L p(J ) (J )wJ 2 p

.

Inserting (J ) = L p+1 (J ) L p 1(J ) in the above equation and performingstraightforward manipulations, we arrive at the conclusion

wJ =2[ p(1 + ) + ]

p( p + 1) L p(J )[L p+1 (J ) L p 1(J )]J = 0 , 1, . . . , p

as required.


26/26


REFERENCES

[1] M. Ainsworth , Discrete dispersion relation for hp -version nite element approximation at high wave number , SIAM J. Numer. Anal., 42 (2004), pp. 553575.

[2] M. Ainsworth and H. A. Wajid , Dispersive and dissipative behaviour of the spectral element method , SIAM J. Numer. Anal., 47 (2009), pp. 39103937.[3] S. Challa , High-order Accurate Spectral Elements for Wave Propagation , Masters thesis in

Mechanical Engineering, Clemson University, Clemson, SC, 1998.[4] G. C. Cohen , Higher-order numerical methods for transient wave equations , Scientic Com-

putation, Springer-Verlag, Berlin, 2002.[5] I. S. Gradshteyn and I. M. Ryzhik , Table of integrals, series, and products , 4th edition

prepared by Ju. V. Geronimus and M. Ju. Cetlin. Translated from the Russian by ScriptaTechnica, Inc. Translation edited by Alan Jeffrey, Academic Press, New York, 1965.

[6] M. N. Guddati and B. Yue , Modied integration rules for reducing dispersion error in niteelement methods , Comput. Methods Appl. Mech. Engrg., 193 (2004), pp. 275287.

[7] M. N. Guddati and B. Yue , Dispersion-reducing nite elements for transient acoustics , J.Acoust. Soc. Am., 118 (2005), pp. 21322141.

[8] F. Ihlenburg , Finite element analysis of acoustic scattering , Applied Mathematical Sciences132, Springer-Verlag, Berlin, 1998.

[9] D. Komatitsch and J. Tromp , Spectral-element simulations of global seismic wavepropagation-I. Validation , Geophys. J. Int., 149 (2002), pp. 390412.

[10] D. Komatitsch and J. Tromp , Spectral-element simulations of global seismic wavepropagation-II. 3-D Models, oceans, rotation, and self-gravitation , Geophys. J. Int., 150(2002), pp. 303318.

[11] D. Komatitsch and J. P. Vilotte , The spectral-element method: An efficient tool to simulatethe seismic response of 2D and 3D geological structures , Bull. Seismol. Soc. Amer., 88(1998), pp. 368392.

[12] F. Maggio and A. Quarteroni , Acoustic wave simulation by spectral methods , East-West J.Numer. Math., 2 (1994), pp. 129150.

[13] K. J. Marfurt , Accuracy of nite-difference and nite-element modeling of the scalar and elastic wave equations , Geophysics, 49 (1984), pp. 533549.[14] A. Ralston , A First Course in Numerical Analysis , McGraw-Hill, New York, 1965.[15] G. Seriani and S. P. Oliveira , Optimal blended spectral-element operators for acoustic wave

modeling , Geophysics, 72 (2007), pp. SM95SM106.[16] L. L. Thompson and P. M. Pinsky , Complex wave number Fourier analysis of the p-version

nite element method , Comput. Mech., 13 (1994), pp. 255275.

Documents

Ainsworth Optimally Blended Spectral Fe Scheme for Wave Propagation and Nonstandard Reduced Integration