Box schemes on quadrilateral meshes

Comput ing 51 ,271-292 (1993) C O I ~ [ ~ [ ~

�9 Springer-Verlag 1993 Printed in Austria

Box Schemes on Quadrilateral Meshes

T. S c h m i d t , Kiel

Received June 23, 1993; revised September 8, 1993

Abstract - - Zusammenfassung

Box Schemes on Quadrilateral Meshes. Box schemes (finite volume methods) are widely used in fluid- dynamics, especially for the solution of conservation laws. In this paper two box-schemes for elliptic equations are analysed with respect to quadrilateral meshes. Using a variational formulation, we gain stability theorems for two different box methods, namely the so-called diagonal boxes and the centre boxes. The analysis is based on an elementwise eigenvalue problem. Stability can only be guaranteed under additional assumptions on the geometry of the quadrilaterals. For the diagonal boxes unsuitable elements can lead to global instabilities. The centre boxes are more robust and differ not so much from the finite element approach. In the stable case, convergence results up to second order are proved with well-known techniques.

AMS Subject Classification: 65N 15, 65N99, 35A40

Key words: Box method, boundary value problem, finite volume method, variational formulation, stability, error bounds.

Box-Methoden auf Vierecksgittern. Box-Methoden (Finite-Volumen-Methoden) sind verbreitete Verfahren zur L6sung physikalischer Erhaltungsgleichungen, insbesondere in der Str6mungsmechanik. In dieser Arbeit werden zwei Methoden ffir elliptische Differentialgleichungen untersucht, die Diagonal- Boxen und die Schwerpunkt-Boxen. Da die Box-Methoden im Sinne yon Petrov-Galerkin-Verfahren interpretiert werden k6nnen, erh~ilt man vergleichbar zur Finiten-Element-Methode eine varia- tionsrechnerische Stabilit~its- und Fehleranalyse. Damit werden O(h)- und O(h2)-Fehlerabsch~itzungen hergeleitet. Lokale Eigenwertprobleme f/ihren zu Stabilit~itsaussagen. Allerdings ergibt sich eine Abh~ingigkeit vonder Anzahl und Art gest6rter Vierecke. Insbesondere die Diagonal-Boxen sind anfiillig ffir lokale St6rungen.

I. Introduct ion

The box method (finite volume method) is a discretization technique for part ial differential equations, which is na tura l for the numerica l solut ion of the conserva- t ion laws, because they generate conservative schemes. Most of the investigations concern fluid dynamics.

In this work we want to determine the solut ion of a b o u n d a r y value problem in some gridpoints of the doma in g2 c ~z. There is a subset called box (cell, control volume) assigned to each gridpoint . The idea of the box method is to guarantee conservat ion over each box. In fact, this results in a k ind of finite difference scheme. In the literature, the ma in interest is given to the so called cell-centre method (e.g.

272 T. Schmidt

see Dick [11]), which is box-oriented in the sense that first the boxes are constructed and then the barycentres of the boxes are chosen as gridpoints. An alternative is the cell-vertex method (see Mackenzie, Morton and Siili [17], [18]) where the gridpoints are the vertices of the boxes, which has the disadvantage, that the number of gridpoints and the number of boxes are different. Another approach, more comparable with the finite element method, is presented by Patankar [3], [19], and called control volume finite element method. This method is characterised by first building a finite element mesh, choosing the element vertices as gridpoints and then constructing boxes around these points. The present work starts from this view point, leading to an analysis of Petrov-Galerkin methods with the standard finite element space as trial space and piecewise constants (over the boxes) as testfunctions. In this sense, Bank and Rose [4], Hackbusch [14], and Cai [7], [8], [9] analysed second order boundary value problems using triangulations, whereas Heinrich [15] prefers the finite difference view point. Especially in the physical applications, one uses quadrilateral meshes. Siili [21], [223 treated quadrilateral schemes for hyper- bolic equations and the cell centre method for Poisson's equation. This was also done by Girault [12]. But they used grids of rectangules or quadrilaterals close to parallelograms as they are often used in the applications. That means that, in general, they deal with affine families of finite elements. We were interested in the isoparametric case, where the transformation is nonlinear so that many new problems occur.

The outline of the paper is as follows: first we introduce the model problem and the notations. We consider two box methods. The first one is named diagonal boxes because the diagonals of the quadrilaterals form the box boundaries. On square grids they look like rhombs. Similar discretizations can be seen in Albrecht [1] or Mackenzie and Morton [17]. This method is of special interest because of a double overlapping of the domain occurs. The second method is the generalization of the cell centre method on general grids, which are called centre boxes. This is a natural approach, similar to triangular box methods. In the main part of this work, we make a stability analysis for both methods, i.e. we prove the inf-sup condition for the arising bilinear forms. This is done by an elementwise approach. The stability depends on the number of quadrilaterals not beeing parallelograms respectivily the strength of the nonlinearity, which for quadrilaterals can be expressed in terms of the distance of midpoints of the two diagonals. The application of the Bramble- Hilbert Lemma [6] or its bilinear versions give the convergence results up to second order. Finally, we mention generalizations and check the involved assumptions in the stability theorems by a numerical test.

2. Preliminaries

2.1. The Modelproblem

Let ~ ~ ~2 be a polygon with boundary F := 0s We consider the strongly elliptic boundary value problem in divergence form:

Box Schemes on Quadrilateral Meshes 273

- d i v ( A V u ) = f in (2 (1)

u = 0 on F,

where A is symmetric. Let amin denote the ellipticity constant, i.e.,

(w,A(x)w)>_ami,[w[ 2 for all x ~Q, w ~ ~ 2 . (2)

If no explicit condition is given, we assume that all coefficients of A, f, and F are smooth enough so that the boundary value problem (1) is equivalent to the following variational formulation:

F i n d u ~ o l ( g 2 ) wi tha(u ,v ) :=(AVu, Vu)o=(f ,V)o for all v e J((ol (~), (3)

with ~o~(s denoting the usual Sobolev space of weakly differentiable functions with zero boundary condition and ( . . . . )o denoting the usual Lz(~Q) inner product.

2.2. The Finite Element Mesh

Let (Th)h> o be a finite element family. For every h, Th is a set of finite elements. Furthermore, we assume that every element is a quadrilateral and that the family is conform, i.e., that the union of all elements covers s but that the intersection of the closure of two elements may only be empty or a common edge or corner.

Notations: For every set T of finite elements and every K ~ T we define:

h K diameter o fK ; (4)

Pr radius of the largest circle contained in K ; (5)

/~r length of the smallest side o fK ; (6)

O r the interior angle of K with the largest absolute cosine; (7)

m K the distance vector of the diagonal midpoints o fK. (8)

The parameter h is defined as h := sup{hr: K ~ Th}.

Definition 2.1. ([5], II.5.1(2), III.2.2) (Th) h >o is called quasiuniform, if there exist Cr, sr > O, s r < 1 independently of h satisfying the followin9 conditions:

(i) Each quadrilateral K ~ T h is convex and (9)

(ii) a) hK/h K < C, (10)

b) ]COS(0K) [ < 1 -- S r. (11)

For the distance mr, the bound [mr] <_ const o hr is obvious. Requiring the strength- ened condition [mr[ < constx h E, S/ill shows in Lemma 1 of [21] that, in the case of quadrilateral meshes affinely equivalent to rectangular meshes, condition (i) is equivalent to the well-known condition hr /pr <- Cr for all K e T h. In [-20] (Lemma 2.2.4) we generalized this result in the way that the equivalence also holds, if const o is small enough but independent of h. The proof is a natural generalization of Siili's proof.

274 T, S c h m i d t

The finite elements in Th generate the gridpoints, namely the vertices of the elements. Similar to Hackbusch [14], we denote the set of vertices by

G(f2) := {P e ~: There exists a K s Th with vertex P}. (12)

Let Go(O) := G(f2)\F, n h the number of interior gridpoints, ad V h := N"". The usual finite element spaces are

5~h(f2) := {u e cg0(~): u o j ; 1 bilinear on E for all K e Th} (13)

with the reference element E := [ - 1, 1] 2. JK is the bilinear transformation from E onto K. Let 5~163 := 5eh(f2) n ,r If we use the usual Galerkin basis

{101P = Q (14) ~be~ := p -# O'

it is plain that dim 5rob(f2 ) = dim Vh = nh and

ze := supp(~b~) = {K ~ Th: P e K}. (15)

The natural Galerkin projection pO: Vh ~ 5eoh(f2) is given by

P~(uh) := Z UeOg, (16) P ~ Go( ~21

2.3. The Bilinear Transformation

For a quadrilateral K with corners P1, Pz, P3, P4 we define the vectors

(P2 - P1) + (P3 - P4) (P3 - P~) + (P, - P2) a : = b : =

2 ' 2 ' (17)

(P1 -- P2) + (e3 - P4) t/'/ : m

2

With respect to the finite element theory, the quadrilaterals are considered as isoparametric elements in the sense of Ciarlet [10]. The bilinear transformation JK: E ~ K is given by

a b m Jr(~,t/) := z + ~r + ~ t / + ~- ~t/, (18)

where z is the barycentre of K. m is the vector describing the distance of the midpoints of the two diagonals of the quadrilateral K, therefore m measures the nonlinearity of the transformation Jr . The determinant of the Jacobian J r is important for transformation properties, We have

det(DJr(~, tl)) = c~ + fl~ + 7tl, with (19)

vol(K) det(a,m) vol(T2) - vol(T1) det(m, b) vol(T4) - vol(T1) ~ - , f l - - - - , y - _ _ -

4 4 4 4 4

Figure 1 illustrates these quantities:


/'4

/'l P1

P1

P1

Figure 1. The bilinear transformation J r and the parameters a, b, and m

For I E N and M c R 2 define:

I[glh, oo, M := sup{lD~g(x)l: c~ multi-index, I~[ = l, x e M}.

With this notat ion, a direct evaluation of the norms yields the following theorem, where the dependence of the quadrilateral on m is outlined, compare Siili 1-21] Lemma 8.

Theorem 2.2. Let ( Th) be a quasiuniform finite element family. Then for every K ~ Th:

(i)

(ii) a)

b)

c)

(iii)

a)

b)

c)

(iv)

a)

b)

Jr is bijectiv,

][Jgllx,o~,E < hr,

I[JK[12,o~,E --< �89 2 ,

l ~ I~ ~ I ~ ~ ' l y~(~)

det(DJK) is linear,

1 2 Ildet(DJ/~)llo,~,~ < ~h~,

Ildet(DJK)[I 1,o~,e < �88

Ildet(DJ~l)Ho,~,K < 4C,Zh~ 2 + sin(0K) -1

with C,, 0 K from (2.1).

3C1, C2 > O, independent of h and K with

IIJ~IlII,| _< C l h K 1 ,

]IJ~III2,~,K < C2h;~2lml.

(20)

(21)

(22)

(23)

(24)

(25)

(26)

(27)

(28)

276 T. Schmidt

3. The Finite Volume Method

3.1. The Box Mesh

For fixed Th, let B e c (2 denote the box assigned to the gridpoint P. Let ~ be the family of all boxes. The construction of the boxes has to satisfy similar conditions as finite elements (Heinrich [15], "secondary network", Hackbusch [14], "box mesh"). They must partition the whole domain f2 and should have a piecewise smooth boundary and be conform in the following sense:

Definition 3.1. ~ is conform, if

(i) P ~ B'e and B e c Ze for all P E G(f2); (29)

(ii) U B e = t ? ; (30) PeG(f2)

(iii) Be c~ Be ~ ~Be for all P, Q ~ G(g2) with P v~ Q. (31)

The box mesh induces a linear space Vo, ~, the span of all characteristic functions Xnp:

V~ := { v ~ L~~ ~VhEVh:v= P ~ ~Go(~) (Vh)eZBp} (32)

The projection ps: Vh --" Vo,~ is given by

v h ~-* ~" (Vh)eZB p. (33) P �9 Go(f2)

There are examples of box meshes which do not satisfy (31) (e.g. Hackbusch [14]). A more general definition is given by:

Definition 3.2. ~ is conform, if

(i) ~ fulfils (29) and (30): (34)

(ii) There exists ~* which fulfils (31), and (35)

a) for all B* e ~* there is a B 6 J): B* ~ B, (36)

b) for all B ~ ~ : ~ B* = B, (37) B*E ~* ,B*~B

(iii) dim Vo, a = nh. (38)

If the box mesh fulfils Definition 3.1, the choice ~ * = ~ proves that Definition 3.2 is valid. The analysis of box methods is based on the bijective relationship Vo,~ := Pn(Vh). With this notations we can define a regular box mesh:

Definition 3.3. M is regular, if

(i) ~ is conform accordino to Definition 3.2, and the finite element family (39)

(ii) T ~ ' : = { B * ~ K : B * ~ * , K ~ T h , vo l (B*nK)>O}isquas iun i form. (40)

Integrating (1) over a box Be and applying the Gaul3ian Divergence Theorem, we get a boundary integral instead of a volume integral:


n p p p

The finite volume method searches for a solution u B satisfying (41) for all P ~ G(s This can be formulated in the following way: One multiplies (1) with a test function v ~ Vo,~, integrates over s and applies formula (41):

P ~ Go(-Q) Bp P e Go(12) e

The discrete solution is the function from ~oh(s satisfying (41) for all v e Vo, ~. We obtain a nonconforming Petrov-Galerkin method, because the trial and test space differ and the test space Vo,~ is not contained in J/(o~ (O). The method results in the linear system

L B - - b(qS~,)~Bp) and fh~ := fB(ZB,). (43) LahUh = fh ~ with hPe . -

In general, Lh 8 is nonsymmetric; in fact in the case of general quadrilateral meshes the matrix of the box method and the finite element Galerkin matrix a(q~, ~ ) differ in contrast to the case of triangular meshes (cf. Hackbusch [14]) so that a new stability analysis is necessary.

3.2. The Diagonal Boxes

Let P e G(fJ) and let K ~ z e be a quadrilateral. The diagonals split K into four triangles. Let Bp,K be the union of the triangles with vertex P. Albrecht [1] uses these boxes to gain a discretization of the diffusive terms of convection-diffusion equations. In the paper of Mackenzie and Morton [17], similar difference stars arise by discretizing the diffusive terms by the cell-vertex method. It is obvious that in this case the boxes are not disjoint, namely if two elements have a common side. t2 is overlapped exactly twice.

Figure 2. Part of the diagonal box Bp in the quadrilateral K

Theorem 3.4. I f z is quasiuniform then ~ is regular.

Proof." Everything follows from the regularity of the finite element mesh and the choice with

278 T. Schmidt

o ~'* := {Bp A Bo: P, Q E G(O), P # Q}.

For a detailed proof see Schmidt [20], Satz 3.2.2. �9

Remark 3.5. For the Neumann Problem there are domains O and meshes G(O) such that the LZ(O)-norm I1" [Io,~ is not a norm for Vo.e.

Proof." (chequerboard error) Let n ~ N, h = 1/n. O = [0, 1] z is divided into

z : = { h [ i - 1, i] x h [ j - l , j ] : l < i , j < _ n } .

Then u h ~ G(O) defined as (Uh)ij :~- ( - - 1) (i+j) fulfils pB(Uh) = O. �9

Because of P~(1) 2, from now on define B 1 = PJnew :-~ ~P~ld without changing the nota- tion. Analogously, we use the bases (�89 P e Go(O)) in Vo, ~.

For P, Q 6 Go(Q), P r Q let 7eQ := OBp ~ OB o. In the interior of O, the unit outward normal vectors np, nQ point into opposite directions: np = -nQ (cf. Patankar [19], 3.4, Rule 1). This gives rise to a semi-norm in Vo,e comparable to the usual Sobolev semi-norm I'11. We introduce the index sets:

~ :-- {{P,Q} c G(O): I~ml > 0}, ~r (44)

~ : = { { P , Q } e ~ C r : l y p o c ~ K l > O } for all K e Th.

Then we have the following Vo, e semi-norm:

2 .__ Q ~ [ull,~ . - ~ (u(P) - u(Q)) 2 := ~ (u(P) - u(Q)) 2, (45) {P,Q}~ ~'r K~ T h k{P, ~/r )

: =]u'(~,~, K

which makes also sense for continuous functions in O or vectors in V h. We require that this is a norm on Vo,~ fulfilling an inequality of the Friedrichs-Poincare type, i.e., [I pB(Uh)II O, o <-- C I PB(uh)I1, ~. This is easy to prove for structured grids but holds even for more general grids. In fact, there is no equivalence of the form: C1 IPB(Uh)]I,~ ---% IIPO(uh)l[ l, o -< C= IP/l(Uh)l:t,~ for all u h e V h with constants independent of h. However, a simple comparison of lul~,~,~: and the usual local Sobolev semi-norm l ull,k shows the first inequality. This implies also an inverse inequality for the norm I" I1,~ (cf. Schmidt [20], Section 3.3).

3.3. The Centre Boxes

Let P e G(O) and K e The a quadrilateral. If one connects the midpoints of the sides of K with its barycentre, K is divided into four smaller quadrilaterals. Bp,~ is the small quadrilateral with vertex P. The centre boxes are defined analogously to the triangular meshes (see Hackbuseh [14], Bank [4], Heinrich [15], Cai [7]) and can be seen as a generalization of the cell centre methods for rectangular grids. A complete analysis for structured rectangular grids is given in the works of Siili [22] or Girault [12].

B o x S c h e m e s o n Q u a d r i l a t e r a l M e s h e s 279

P W w

P w w

F i g u r e 3. P a r t o f t he c e n t r e b o x B e in t he q u a d r i l a t e r a l K

Theorem 3.6. I f z is quasiuniform then ~ is regular.

Proof" Everything follows of the quasiuniform finite elements, pB is bijective because the boxes are disjoint. �9

The semi-norm on V h is similar to the case (45) from above:

2 lTPe n KI luh,~ := 2 (u(P) - u(Q)) 2 := 2 ,~;~ { v , e ~ o r~rh {l,,,~l~r~ [~vel (u(P) - - u(O)) 2. (46)

t ) Y_

An elementwise eigenvalue analysis shows the equivalence of the norms [IP~(')[Io, with IIPG(-)IIo,,~ and IPB(.)h,~ with IP~(.)h,~ on Vh, where the constants are independent of h.

4. Stability Analysis

Unfortunately, the bilinear form b from (42) is not positive definite for all meshes. A more general stability theorem is given by Babu~ka I-2]: b has to be continuous and has to fulfil the well-known inf-sup condition:

inf sup Ib(u,v)l > C. (47) u ~ s%h(12), lu[1.~ =i v E Vo,~, Ivh,~ =i

The constant C should also be independent of h. Our approach (cf. Bank and Rose [4], Liebau [16]) is to write b as a sum of all elements K e T h. For every K, we derive a matrix representation with a 4 x 4 matrix. Then we investigate the symmetric parts of these matrices in the hope that most of them are positive definite. The bilinear form b can be written as the following sum:

P e G ( ~ 2 ) B e \ l ~

280 T. Schmidt

K~Th {P, ~R ~pQt~K)\F

= Y', <Brur, vK>. (48) K~ T h

To simplify the notations we assume that the matrix A is piecewise constant over all K e T h

4.1. Stability of the Diagonal Boxes

In the case of the diagonal boxes, the local 4 x 4 matrices BK can be computed with the help of the 2 x 2 identity 12 as B r = (I2, -I2)t(/3r), where the 2 x 4 matrix/3K is defined by:

1 f~ (ni, AVq~>ds. (49)

For the definition of the vectors a, b, m compare Fig. 1. Let 7~ be a parametrization of the diagonal (b - a) (resp. 72 of - ( b + a)), let n~ := (7'i)~ the outward normal vector (with (p,q)~ := (q , -p) ) and ~ff ~ 6ah(f2) the basis function associated to Pj E K. ?i := j{1 o 71 is the transformed piece of the boundary.

f f (rlb ~ - r176 - a)~ rl := - 8 d ~ , ~ - ) ~ R~: := (0,0, rl - -rz , r 1 +r2)

f [ <rlb ~ - r176 + a)~ r2 := 8 det DJr(~, rl)

o ~2(t) dt,

(5O)

(0, 1 det(a, m) det(m, b)~ Sk : =

\ ' vo l (K) ' vo l (K)J (51)

1 We introduce ~ := A1/2a ~ b := A1/2b~ With T[] = - 1 1 it

1 - 1

- 1 - 1

i 0 0 0 T:z B K T ~ - 1 0 0 0

vol(K) 0 lal 2 - <a, ~>

o - < a , g > Igl 2

follows that

i ) -.r

: =Bo ,~

+ R,,Sk. (52)

Example 4.1. Let n e N, h = 1/n. If we discretize the Laplacian equation on Q = [0, 1] 2 using the grid Th := { h [ i - 1,i] x h [ j - 1,j]: 1 < i,j < n}, we get the difference star:

Box Schemes on Quadrilateral Meshes 281 [10 1 0 4 (53)

- 1 0 -

It is interesting that the finite element method with 1-point quadrature results in the same scheme (cf. Ciarlet [10], Exercise 4.1.8 (iii)), i.e.:

f r ( V v , Vu) ~ vol(K)(Vv, Vu) o Jr(O, 0). dx

The following lemma shows that the perturbation RKS ~ in (52) depends on mr; especially for parallelograms, it vanishes.

Lemma 4.2. There exists a constant C > 0 independent of h with [Rr] <_ C [m[ . hr

Proof." Using the trapezoidal quadrature rule to approximate ri , we get:

det(m,a + b)(b + a,A(b - a)) amaxC]lm[ rl ~ 8(det D Jr(l , - 1))(det D J r ( - 1, 1)) < s~h r

The quadrature error of the trapezoidal rule is fixed by the second derivative of the integrand I in (50), which has the form I = (z/det DJK) o ~1, where z is linear, so that

I"(t) = ([(De(z/d)) o ~ (t)]~] (t), ~] (t)) + ( IV(z/d)] o ~'l(t), ~ (t)).

From the transformation rules (Theorem 2.2) it follows that the occurring first derivatives are uniformly bounded by a constant independent of h and all second derivatives include a partial derivative of det DJ K as a factor, which guarantees bounds by [mr[. �9

Unfortunately, the norm 1.[1,~ is not equivalent to the Sobolev norm [. h,o. For quadrilateral grids consisting only of parallelograms this does not matter, since the above perturbation vanishes so that the positive semi-definiteness of the matrix Bo,r implies continuity and ellipticity of b, if we use the norm 1. [a,~ for trial and test space. For general meshes, the perturbation is not zero so that negative eigenvalues occur. For a stability result by this local analysis, we need conditions to restrict the number of such elements. On 6~oh(s we define the norm (I'I~,K is the ~ i ( K ) semi-norm)

[U[2,~r t2:= ]PB(pG)-I(u)[2,~ "]- E ----(linK[j2 r~r, \ hK-J [u121'r" (54)

Theorem 4.3. Let ~ be the diagonal boxes. Let Co, Ca, C2, Ca > 0 be constants independent of h and T O := {K ~ Th: Im~l > Coh~}, 7"1 := {K ~ To: Im,d < C~h~/2} and T 2 := To\T I. Assume that IZll -< C2 h-~, IT21 - C3.

Then there are constants Co, C1, C2, > 0 independent of h such that

I I1 ,~ , E 2 _ - luh ,K < C / l u h , ~ K~Th\To \ hr ) lUI~,K < (~o U 2 r~r, \-~-K J (i = 1, 2).

(55)

282 T. Schmidt

i.e., the norms I"II,.,~.Q and [PB((P ~ 1('))J1,~ are equivalent on ~g(f2) and b is continuous:

Ib(u,v)l < CslPB((P~ for all u e 3~g(f2), v e Vo,xe. (56)

I f the constants Ci, i = l, 2, 3, are small enough then there is a C > 0 independent of h with

inf sup Ib(u,v)l >_ C. (57) ue ,7~Oh(g2),IPB((P% lu)h..m=l veVo,~,lvh,~=l

Proof: The inequalities (55) immediately follow, because for i = 1, 2 the Euclidian norms of the form ~,~K,K~ r, u2 can be bounded by [u12~ under consideration of the number of elements from Tv From (54) we have 1' [1,B,o > [PB(PG)-I(')[I,~' (55) guarantees the inverse direction, i.e. both norms are equivalent. Let u s 5gob(O) and v e Vo,~. From (52) it follows that

(BKUK, VK) = TDUK, Bo,KT~VK) + (TDUK, RKStKTDVK).

The matrix Bo,K is positive semi-definite. An easy computation shows that the smallest positive eigenvalue is bounded from below by (area x is the largest eigenvalue of A)

1~/[2 [/~[2 det(8, b) 2 det(A1/2(a,b)) 2 det(A) vol(K)

vol(K)( la l 2 + I/~l 2) vol (K)( l~l 2 + I/~l 2) vol (K)( l~l 2 + I{,I 2) I~l 2 + 10[ 2

aminlallblsr amlnSr aminSr ~o. >_ > am.x(lal 2 + Ibl 2) {lal [bl'~ amax(1 "[- Fir)

With T~ from (52) we can write:

[ ( BK TDUK, TDV~ ) [ < I ( B1/,2K TGUK, B~/,ZK TDVK ) + I ( SK, T~UK ) (RK, T~vi~ ) l

<@~lul,,~.,,+C~lh;lluh.K)lvl,,~,,K. The Cauchy-Schwarz inequality implies the continuity of b. We choose v = PB(P~ Because of v(P)= u(P)/2, it follows that lul~,e. Using the smallest eigenvalue calculated above and the same bound for the perturbation term, we derive:

b(u,v) >_ eolull,~- C1 K~Th [ul2"K ]V]l'~"

Now (57) is valid if the above constants are small enough. I

4.2. Stability of the Centre Boxes

The advantage in this stability analysis is that there is an equivalence of the ~o~ (O) semi-norm with l" 11,~- Therefore condition (47) holds for more general grids, only extremely distorted quadrilaterals (i.e. ]linK[ [ ~ hK) must be neglected. Unfortu-


nately, this local analysis considers the worst case over all elements. A global technique taking averaging effects into account is not available The size of the cons tant C , in the next Theo rem is quantified in the proof.

Theorem 4.4. Let ~ be the centre boxes. Then the bilinear form b is continuous and there are C > 0, C , ~ (0, 1), all independent of h so that, if for all K ~ T h ImKI < C, hK is valid, the following stability estimate holds:

inf sup [b(u,v)[ > C. (58) u~ 5eoh(f2),juh,a =1 v ~ Vo,,e =1

Proof." The continuity follows as in the p roof of Theorem 4.3. Let u ~ ~Oh(t2) and set v = pB(pG-1 (u)). A decompos i t ion of b as a sum like in (52) gives local matrices Br of the form

TDBK T~ = Bo,r + diag(O,#l,t~2,t~3)SrStr (59)

f~ ~tl~lz f~ ~tl~124(ct 2 _ (60) with Pl := 4(~Z~-fl)2)dt + (t~)2) dt

f f cq2lal 2 f f ~t2l/~' 2 �9 and , 2 := 4 ( ~ - - - ~ ) 2 ) d t , , 3 := 4 ( j ~ (~-7)2) at-

With the nota t ions d r := 1/2 min { #2, #3 }, d i := #~ - dK, D r := diag(0,/11,/~2,/13) , and D := diag(O, dx,dE,d3), ur := T~ur we get

(BKUr, ur) = (UK, Borfir) + (uK, DKSrStrfir)

= (~K, BoK~K) + (fir,�89 + SKSkDK)~K>

= (~K,(BoK + drS~cS~)~r) + (~K,�89 + S~S~D)~r).

An elementary compu ta t i on shows

f~ I~b- ~1 ~ lul~,K = (TDUK,(BoK + #SKS~)TDuK~, # := ~ 16detDJr(~,rl)d(~,rl).

~3

=~ (r~uK,(Bo~ + dKSKS~)T~ur) >_ min 1, lul~,r.

�9 + S O, soft e orm'= (: 0). The characterist ic po lynomia l pS of the symmetr ic 3 x 3 matr ix S is pS(2 )= -2(2 2 - p2 - q) with coefficients:

~ ( d , - d~) ~ ~,~(d~ - da) ~ Bg~(d~ - d~) ~ p := trace(S) >_ d~ > 0 and q : - 4e 2 + 4c~2 + 4c~

The inclusion 0 < q < C holds. S has only one negative eigenvalue 2_

bounded f rom below by

284 T. S c h m i d t

p p/~T2 2q r 2_ - 2 X / 4 + q > - - - > - C ~ - p - \ h K J "

(TDUK, ST~UK) > - C ^ 2 L/I1,K �9 - lu l l ,== ~ - - 1 L hK .,] I 2

~ ( B r u r , uK) >_min 1, lul~,~-C~\~-K ) lul~,K.

The regularity of ~ implies that dK/# is uniformly bounded from below by C o > 0. ImKI

Choose C, < (min{1,dK/g}) m. Then ~ _< C. implies, because of lul~,e -- Ivh,:,~,

that

Ib(u,v)t > min{am~,,Co/2}lul2 a >_ C2lUll,alUll,~ = C2lul~,alvl~,~lvlx,~. [ ]

5. Convergence Results

Let H: ~go(~) ~ yh((2) denote the interpolation operator. We reduce the estimation of the consistence error to the problem of estimating the interpolation error u - H(u) (cf. Ciarlet [101 or Cai [71). Braess ([5], Satz 6.7) shows that for parallelogram meshes the (bilinear) interpolation error is bounded:

HU-- Hu[im, a <-- C h 2 - m l u l 2 , o for all u ~ Yf2((2), m = 0, 1,2. (61)

This result can be generalized to higher order, if u is smooth enough. For general grids, (61) can be proved for m = 1 if one uses the special semi-norm ['12,e on the reference set E, the usual Jtf2(E) semi-norm minus the mixed derivative D ~1' ~) (cf. Ciarlet [101, Exercise 4.3.9 or Schmidt [201, Appendix B).

Let H~: ~o (~ []) ~ Q 1 (E ~) be the interpolation operator belonging to the reference set. Let K e Th. For all c e ~o(~) we have: (Hc) o JK is the bilinear function on E with the same values at the corners of E as c o JK, i.e.,

(~)(c o JK) = (He) o JK. (62)

Let {P,Q} ~ ~K and V = VpQ c~ K. With the notations '3 = J~1(7), A := A o JK, and = v o JK we define the following functionals:

r ~2(K) ~ [~, V ~ --f~ (n, AV(v -- Hv))dF; (63)

~r: ~2(E)--+ N, ~--~ - f , @, ,4DJ~W(O- f l O ) ) ~ d f ' . (64)

Lemma flA. ~pr(v) = ~br03) and ~(~) = 0 hold, i f ~ is bilinear on E.

Proof: The first equality follows by a simple computation, the second one from sT,~ = ,~. �9


Lemma 5.2. Let K �9 T h be a parallelogram and A be a constant on K. In the case o f the diagonal boxes, we have (or(O) = 0 for all f �9 ~2(E []).

Proof." Lemma 5.1 implies the assertion, provided that f e QI(E[]). f(4,r/)= 42 implies that H f - 1 . It follows that V ( f - / ) f ) = (24,0) so that (Or(O)= ~11 const ~ d4 = 0, because ,4, D Jr , and p are constant. The case f((, r/) = r/2 can be treated analogously. �9

In the case of the centre boxes one has a similar result under additional conditions. This is proved by Siili ([22], Theorem 4.1) for rectangular grids.

Lemma 5.3. Let {P, Q} �9 ~ , K l, g 2 E ~C with ?eQ c K : = K 1 ~d K 2. I f K, K1, K 2

are parallelograms and A is constant on K, it follows that (Or(O) = O for all f �9 ~2(E ~).

Proof." In this case, the transformation Jr is an affine mapping. Let us assume that K 1 is affinely transformed onto E 1 := [ - 1, 1] x [0, 1] and K 2 onto E 2 : =

[ - 1, 1] x [ - 1, 0]. Then y is just a parametrization of the connecting line of the midpoints of E 1 and E 2. The other possible cases are proven analogously./)f is the piecewise bilinear interpolation from f = v o J r (on E l, E2). Nothing is to do if f is bilinear on E. f(4, q) = 4 2 implies fff = (24, 0) t , / I f = 1 on E i and on E 2. This shows that (Or(0) = ~-/2/2 eonst ~ d( = 0. If f(~, ~/) = ~/2 then fff = (0, 2~/) '. Therefore we have /Tf = ~/on Ea but Hb = - ~ / o n E2. It follows:

2q dq - l dq + l dq = 0 . �9 = c o n s t [

Lemmata 5.1, 5.2, and 5.3 describe the polynomials lying in the kernel of the functionals. A combination of Theorem 2.2 and the interpolation results gives:

Theorem 5.4. Let the boxes be constructed as in Definitions 3.2 or 3.3.

(i) There is a Ci > 0 with [tPr(v)l < Cih~;[vl2,K for all v �9 jt~ (65) (ii) Under the assumptions o f Lemma 5.2, there exists a C2 > 0 with

]qK(V)I -< C z h 2 j v [ a , K for all v �9 ~:~3(K) . (66)

(iii) Under the assumptions of Lemma 5.3 there exists C3 > 0 with

[ ~ o r ( v ) ) < C 3 h Z ] v ] 3 , r f o r a l l v e ~ 3 ( K ) , w h e r e K = K l w K 2 . (67)

The constants C~, C 2, C 3 are independent of h~, hr, Or, and are uniformly bounded by a constant Ce, ~ depending only on C,, s,, and I[ A H ~, ~.

Proof." Transformation rules and the Sobolev imbedding theorem guarantee that

IcpK(v)l = I (o~( f ) l <- Csllf - / l f l l2,E ( * )

with Cs independent of h r , / ~ , 0K. Bilinear interpolation reasons

[q~K(v)[ = I(o,,(f)l s c E f ] 2 , ~ .

Assertion (i) follows by backtransformation. Inequality (,) also implies that

I(oK(f)l S Ca IJflla,E.

286 T. Schmidt

Again the Bramble-Hi lber t L e m m a ([6]) and the affine t ransformat ion prove the vaidity of (ii) and (iii). �9

Unde r the stability assumpt ion these consistency results give the next convergence theory.

Theorem 5.5. Let us be the solution of( l ) . Under the assumptions of Theorem 4.3 and if u~ ~ ~2(~e'~) ~ ~01 (~r'2) it follows for the diagonal boxes ~ that the finite volume solution u B ~ 5P~(f2) fulfils:

(i) [u s - u n l l , ~ <- Clhllu~ll2,~. (68) (ii) I f u s ~ #g3((2) c~ ~ (f2) and all quadrilaterals are parallelograms then

lus - U~ll,~ < CzhZllu~ll3,o, (69)

where both constants are independent of h.

Proof" u~ is cont inuous on ~. Let u I := Hu~ e 6eoh(f2). Because of u~ ~ ~ot (12), u~ 5aoh(s holds. Define e := u~ - uB, e1 := us - uI, and e n := u I - us. Since

b(us, v )=f~(v ) f o r a l l v e V o , e and b(un, v )= fn (v ) f o r a l l v ~ V o , e ,

b(en, v ) = -b(e~,v) holds for all v e Vo,e. u(P)= u~(P) for all P c G(g2) implies le l l ,~ -- leBIl,~. For e B = 0 all is done. In the case of e n r 0 there exists a v

e iv l l ,~ le , ll,~ -< Ib(eB, v)= [b(e~,v)l,

where z denotes the stability constant. Using the Cauchy-Schwarz inequali ty and the index set f rom (44) we can est imate

I )

Every te rm of the last bound can be treated with the consistency results and we get the assert ions (i) and (ii) under suitable assumptions. �9

Corol lary 5.6. Let ~ be the centre boxes. Under the assumptions of Theorem 4.4 the finite volume solution uB ~ 6Cob(f2) fulfils:

(i) There exits a C > 0 independent of h with lus - U~ll,~ -< Chlusl2,a. (70) (ii) I f u s ~ ~3( f2 ) n ~ o 1 (O) and all quadrilaterals are parallelograms like in Lemma

5.3, then there exists a C > 0 independent of h with lus - un]l,~ < Ch2llu~][3,~. (71)

Proof." With the nota t ions of Theorem 5.5 we have: J e l l , a = l e 1 + e ~ l l , a < [ei[1, a + [enl 1, ~- Thanks to the Bramble-Hi lber t L e m m a ([6-1) the in terpola t ion error ]exll,,2 can be bounded by Ch l usl2, o. l eB[1, ~ is bounded analogously to the


proof of Theorem 5.5. Under the assumptions of (ii), we have [e]l,~ = ]en[1,~, so that again the proof from above shows the assertion. �9

6. Generalizations

6.1. Mixed Meshes

If one uses mixed meshes, i.e., quadrilaterals and triangles, the part of a box in a triangle K is constructed as follows: For diagonal boxes we first draw the medians in K. They split K into six triangles. Again Bp.K is defined as the union of the small triangles with vertex P and the whole box Bp is the union of all B~,K. It is obvious that the double overlapping of g2 is preserved.

W W W W W . . . . . . . . . . . . . . W

Figure 4. Part of the diagonal box Bp in the triangle K

w W w ~ w

Figure 5. Part of the centre box Bp in the triangle K

In the case of the centre boxes we use the boxes like Hackbusch [14]. This gives disjoint boxes Bp = U {Bp,K: K c z~}.

In this case the stability analysis is very simple, because the following identity holds (Hackbusch [14], Proposition 3.1.2):

(BKUK' VK) = fK (AFu, Vv)dx.

288 T. Schmidt

6.2. Mixed Boundary Conditions

In Schmidt [20], we use mixed boundary conditions. In this case, one gets a new bilinear form b, which includes a boundary integral over that boundary part of/7, where the mixed condition is prescribed (see Heinrich [15]). Neumann boundary parts only lead to a modified right hand side (see Hackbusch [14] or Heinrich [15]). The treatment of the bilinear transformation is easier, because a function on 5eh(f2) is linear on element boundaries. Again the diagonal boxes have problems with the chequerboard functions (cf. Example 3.5).

6.3. General Coefficients

If we use variable coefficients like in Heinrich [15], the analysis is nearly the same. One has to assume that these coefficients are piecewise smooth enough, more precisely, the difference of a mean value As and the values of A on K must be bounded by some Ch p (see Schmidt [20]).

6.4. Quadrature Formulas

A similar analysis like in Ciarlet [10] (chapter 4.1) or in Cai [7] gives sufficient results for using quadrature rules for the occuring integrals. Our stability analysis can be used to derive an inequality comparable to the Strang-Lemma and then to estimate the occuring terms with the Bramble-Hilbert Lemma [6]. To prove stability for the diagonal boxes, it is necessary to use more than one quadrature point in the boundary integral occuring in (41). Simpson's rule has the advantage that only one nonlinear system must be solved since two quadrature points are gridpoints. A two-point GauB formula leads to two nonlinear systems but needs less evaluations of coefficient functions. For the integrations in the case of the centre boxes, piecewise one-point rules are sufficient. The volume integrals on the right hand side can be evaluated by one-point rules, but in general they must be split into smaller domains D i, so that the Di c~ K are disjoint. For details see Schmidt [20].

7. Numerical Tests

To underline the assumptions in the stability results (Lemmata 4.3, 4.4), we made some numerical calculations for the Laplace equation - A u = 0 in f2 = [0, 1] 2 with zero boundary conditions. Our stability proof can be seen as an elementwise generalized eigenvalue problem of the form: For every K ~ Th find a pair (2r, xr) with

�89 + (BK)t)XK = 2~cMrxr, (72)

where �89 + (Br) ~) is the symmetric part of the local discretisation matrix Br ((52), (59)) and M r is the local mass matrix of K. 2r denotes the eigenvalue associated to


the eigenvector XK. Stability holds, if the smallest eigenvalue '~min = infK ~ r,~ 2K has a positive lower bound independent of h. However, 2min is only a lower bound for the smallest eigenvalue 2hml, of the global matrix:

Find (2 ~', Uh) with: 1 g(L h -.b (LBh )t)Uh = ,~hmhu h (73)

with Lh ~ from (43) and global mass matrix M h. Possibly, h 2min has a positive lower bound independent of h different from 2m~,, e.g. local unfavourable effect are smoothed globally. In different experiments for various meshes T h with unfavourable

2m~ n approximately. In fact we solve the elements K we calculated the eigenvalue h following eigenvalue problem, which is equivalent to (73):

Find (,~h, Uh) with: �89 + (L~h)t)(Mh) 1/2U h = )LhUh, (74)

where we approximate M h by a lumped diagonal matrix AI h (see Hackbusch [14]).

The unfavourable elements are constructed as in Fig. 6.

(o,ol diagonal box cenlm box distorted grid

Figure 6. Quadrilateral with large ]ml and associated boxes, respectively grids

Pml Such a quadrilateral fulfils ~ = 0.5 - 1/(4r) and in the case of the diagonal boxes,

guarantees local negative eigenvalues. We use a finite element mesh T h like the right one in Fig. 6, where the number of unfavourable quadrilaterals is proportional to the number of gridpoints. In the columns of Table 1 we increase the perturbation (r: 0.5 ~ 0.72), so that the angle e (Fig. 6) converges to re. Rowwise we halve h. This

h/ mr makes sure that the ratio from / hK will be sufficiently small. For all methods the

entries h 2rain in the first column of Table 1 convergences quadratically to 2~ 2. This value is expected because of the symmetry of Lf, the consistency of all discretizations and because 2~ 2 is the smallest eigenvalue (Hackbusch [13], Lemma 4. 4. 2 for h ~ 0). These results also holds for the finite element results (Table 1). The box methods lose symmetry if Iml r 0 so that no convergence against 2~ 2 can be expected. The results for the centre boxes are nearly as good as for the finite element method. Larger perturbations only lead to small deviations and the eigenvalues are clearly positive. The diagonal boxes show an unstable behaviour. This demonstrates that the method depends more strongly on the local effects, indicating that the stronger assumptions of Theorem 5.5 are necessary. Table 2 shows this behaviour

290 T. Schmidt

Table 1. 2mi . h for gr ids like in Fig. 6

D i a g o n a l boxes

pa rame te r r = 0.5 r = 0.502 r = 0.52 r = 0.65 r = 0.72 # points

h = 1/8 18.7454 18.7452 18.7379 18.2603 17.0454 49

h = 1/16 19.4871 19.4869 19.4876 17.4844 - 1.4748 225

h = 1/32 19.6760 19.6751 19.5895 10.2620 6.8309 961

h = 1/64 19.7234 19.8179 19,3767 - 13,8625 10.7572 3969

Cent re boxes

r = 0 . 5

h = 1/64

r = 0.502 r = 0.52 r = 0.65 r = 0.72 pa ramete r

h = 1/8 19.1611 19.1164 19.1152 19.0473 18.9646 49

h = 1/16 1915814 19.5814 19.5806 19.5273 19.4530 225

h = 1/32 19.6997 19.6997 19.6989 19.6470 19.5722 961

19.7296 19.7294 19.7285 19.6761 19.6002 3969

# poin ts

Fini te e lement m e t h o d

pa rame te r r = 0.5 r = 0.502 r = 0.52 r = 0.65 r = 0.72 # poin ts

h - 1/8 18,9926 18.9926 18.9916 18.9462 18,9153 49

h = 1/16 19.5501 19.5502 19.5497 19.5390 19.5315 225

h = 1/32 19.6919 19.6918 19.6918 19.6891 19.6872 961

h = 1/64 19.7274 19.7274 19.7274 19.7268 19.7264 3969

/ ~ Iml = follows from for various ratios h . The parameters are chosen so that ~-K 2/3

h = 1/8 if [ml =~ 0. In fact, there are grids, which have only few unfavourable elements but show an unstable behaviour for diagonal boxes like in Table 2 (right) for a square grid except one, respectively nine elements.

As a conclusion of these numerical test, one sees that the centre boxes lead to robust

schemes but the diagonal boxes are susceptible to perturbations of ~ J . They are

sufficiently stable for 'normal' grids only. Here it should be noticed that the behaviour of discretizations of the Laplace operator for general grids is equivalent to a transformed operator on a simple grid, for example strongly anisotropic grids.

Box Schemes on Quadrilateral Meshes

Table 2. ).hmi n in dependence of ImKI resp. of the number of unfavourable elements hK

291

Diagonal boxes

lmKI - 0 2 h

hK

h = 1/8 18.7454 18.1158 18.0920 18.1158 17.6089 14.7780

h = 1/16 19.4871 19.3298 18.9705 14.8831 17.5673 4.7727

h = 1/32 19.6760 19.4391 18.4167 5.4765 17.2927 -11.9634

h = 1/64 19.7234 19.5261 1 7 . 5 6 9 7 -7.9877 17.0586 19.7262

1 = # dis. K = 9 # points

49

225

961

3969

8. Conclusion

In this p a p e r we h a v e ana ly sed t w o b o x m e t h o d s . T h e s tab i l i ty ana lys is a n d the

n u m e r i c a l e x p e r i m e n t s p r o v e the i r appl icabi l i ty . T h e cen t re boxes l ead to a nea r ly

o p t i m a l m e t h o d as the f ini te e l e m e n t m e t h o d a n d g u a r a n t e e s s tabi l i ty a lso in the

case o f loca l p e r t u r b a t i o n s . Th is is in fact n o t t rue for the d i a g o n a l boxes , wh ich

h o w e v e r m a y be a s imple a l t e rna t i ve on r egu la r meshes .

Acknowledgements

The author thanks Prof. Dr. W. Hackbusch for his interest in this work and many fruitful discussions. I am greatfully acknowledged to the German Research Foundation (DFG) for supporting the project.

References

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292 T. Schmidt: Box Schemes on Quadrilateral Meshes

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241-260 (1991). [19] Patankar, S.: Numerical heat transfer and fluid flow. New York: Hemisphere 1980. [20] Schmidt, T.: Analyse zweier Finite-Volumen Methoden ftir elliptische partielle Differen-

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Box schemes on quadrilateral meshes

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