ROB STEVENSON (\non-overlapping") errors in each subspace, that is, the errors that are not contained in the subspaces corresponding to coarser levels

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1 Report No. 9533, University of Nijmegen. Submitted to Numer. Math. A ROBUST HIERARCHICAL BASIS PRECONDITIONER ON GENERAL MESHES ROB STEVENSON Abstract. In this paper, we introduce a multi-level direct sum space decomposition of general, possibly locally rened linear or multi-linear nite element spaces. In contrast to the well-nown BP and hierarchical basis preconditioners, the corresponding additive Schwarz preconditioner will be robust for a class of singularly perturbed elliptic boundary value problems. Important for an ecient implementation is that stable bases of the subspaces dening our decomposition, consisting of functions having small supports can be easily constructed.. Bacground and motivation This paper deals with additive Schwarz multi-level preconditioners for solving symmetric second order linear elliptic boundary value problems (cf. [u9, Yse93, GO95a]). We assume a nested sequence of linear or multi-linear nite element spaces M 0 M : : : M J : : : and discretize the boundary value problem on M J using Galerin's method. Additive Schwarz preconditioners are based on a subspace decomposition M J = P H. The corresponding preconditioner can be described by the following steps: For all, project the right-hand side (or current defect) onto H ; solve the reduced boundary value problem on H approximately, that is, e.g. apply a point solver as one step of Richardson or Jacobi iteration; nally, sum the obtained approximate solutions from all spaces H. Well-nown examples of additive Schwarz preconditioners of multi-level type are the BP-preconditioner ([BP90]) and the hierarchical basis preconditioner ([Yse8]). The BP-preconditioner is based on the overlapping decomposition M J = P J M. Due to the overlap, the subspace solvers should be chosen carefully. With exact subspace solvers, errors in \ J M = M 0 would be corrected (J + ) times, whereas errors in M J that are orthogonal with respect to the energy scalar product to M J are corrected only once. This would result in a (spectral) condition number of the preconditioned system that is at least of order J +. In the actual BP-preconditioner, Richardson iteration is applied as a subspace solver. For Poisson-lie problems, this iteration only corrects the oscillating Date. September Mathematics Subject Classication. Primary 5N55, 5N30. Key words and phrases. Finite elements, hierarchical basis, additive Schwarz preconditioner, subspace decomposition, robustness, anisotropic equation.

2 ROB STEVENSON (\non-overlapping") errors in each subspace, that is, the errors that are not contained in the subspaces corresponding to coarser levels. It can be proved that as a result is bounded uniformly in J. The reasoning that Richardson iteration only corrects oscillating errors is however not applicable for zeroth order problems. This explains why the BP-preconditioner is not robust for singularly perturbed operators lie + I ( > 0). Yet, it can be shown that = O(J + ) uniformly in. Adaptations of the BP-preconditioner aiming at full robustness for this type of singular perturbation problems are discussed in [Bor9, GO95b]. As we have seen, redundancy in the subspace decomposition can have certain drawbacs. The hierarchical basis preconditioner is based on the direct sum decomposition M J = L J (I I )M J, where I : M J! M is the nodal value interpolant (I := 0). A direct sum decomposition yields an optimum preconditioner only if the decomposition is (equivalent to) a decomposition that is orthogonal with respect to the energy scalar product. For Poisson-lie problems in more than one dimension, the hierarchical decomposition does not have this property. Yet, assuming exact subspace solvers, it can be proved that (J + ) in two dimensions, so in that case the preconditioner is close to optimal, but in more dimensions grows exponentially as a function of J. The reduced boundary value problems on the subspaces are well-conditioned, and so results for exact subspace solvers carry over to the practical preconditioner based on simple point solvers. For zeroth order problems, the hierarchical basis preconditioner gives exponentially growing condition numbers and so clearly this preconditioner is not robust for + I. A promising direct sum decomposition is given by M J = L J W, where W = M L M (W 0 := M 0 ). This decomposition is not only L -orthogonal but it is also equivalent to an H -orthogonal decomposition (cf. [Osw9, Zha9, DK9, BY93, u9, Osw9, Wan9]). Furthermore, since the reduced boundary value problems on the subspaces H appear to be well-conditioned, application of point solvers with respect to some bases of these subspaces would result into an optimum preconditioner for problems lie + I even uniformly in (robustness), in case these bases are uniformly stable. For one-dimensional linear nite element spaces on a regular grids, such L -stable bases are given by a set of prewavelets. Tensor products of these bases give L -stable bases of multi-linear nite element spaces on regular grids. However, on general nite element spaces suitable bases are not easily nd. For linear nite element spaces in two dimensions on regular grids L -stable prewavelet bases are constructed in [KO95] (cf. [Jun9] for the bilinear case). However, each basis function of W is a linear combination of not less than ten nodal basis functions of M, which maes the preconditioner expensive to implement. Subject of this paper is an additive Schwarz multi-level preconditioner based on the direct sum decomposition M J = L J V, where V = M (;)M M (V 0 := M 0 ) and (; ) M is a discrete but L -equivalent scalar product on M. This decomposition is similar to the L -orthogonal decomposition, but in contrast to that decomposition, on general, possibly locally rened nite element spaces, L -stable bases of the subspaces V are easily constructed consisting of basis functions that are linear combinations of, in case of linear nite elements only three nodal basis functions on that level. We do not expect

3 A ROBUST HIERARCHICAL BASIS PRECONDITIONER ON GENERAL MESHES 3 that there exists an L -stable hierarchical basis of M J consisting of basis functions that are linear combinations of a smaller number of nodal basis functions. Note that the spaces V are not mutually orthogonal with respect to one scalar product, which complicates the development of theory. We will prove that if our decomposition is equivalent to an L -orthogonal decomposition then it is also equivalent to an H -orthogonal decomposition, which means that then the corresponding preconditioner based on point solvers is optimal for second and zeroth order elliptic problems and thus also that it is robust for problems lie + I. For model cases of nite element spaces on quasi-uniform two-dimensional meshes, we are able to prove this L -stability. Our numerical experiments with locally rened nite element spaces show L -, and thus H -stability also in that case. We obtain similar results for an example with a discontinuous coecient, where we apply discrete scalar products that are equivalent to the resulting weighted L -scalar product. Although not subject of this paper, we mention here that our space decomposition also will be useful for solving general H s -elliptic problems for s [0; ] as integral or pseudodierential equations. We refer to [BCR9, DKPS9, DKPS]. Another direct sum multi-level decomposition close to the L -orthogonal one is discussed in [VW95]. In that paper, from each standard hierarchical basis function from (I I )M J an approximate L -orthogonal projection onto M is subtracted. There is a tradeo between the accuracy of this approximation and the size of the supports of the resulting basis functions. At least theoretically, to get a uniformly bounded condition number for Poisson-lie problems, the approximate L -projection has to converge to the exact one if the level tends to innity. Anisotropic equations. The idea of a space decomposition based on orthogonality with respect to discrete scalar products was introduced by Hacbusch in the development of the Frequency Decomposition Multi-Level Method ([Hac89, Hac9]) aiming at robustness for anisotropic operators P d i= a i, where the a i are non-negative but dier possibly largely in size. Both the BP- as the hierarchical basis preconditioner show exponentially growing condition numbers applied to these operators in case of strong anisotropy. A robust BP-preconditioner can be obtained when the coarser nite element spaces are constructed by means of semi-coarsening in directions guided by the relative sizes of the a i ([TCJK9, GO95b]). This approach is necessarily restricted to a i that are essentially constants. For multi-linear nite element tensor product spaces on regular grids, in [Ste9, Ste95] it was proved that our decomposition M J = L J V is equivalent, uniformly in the size of the anisotropy, to a decomposition that is orthogonal with respect to the energy scalar product related to the anisotropic boundary value problem. However, the reduced boundary value problems on the subspaces V are not uniformly well-conditioned with respect to the anisotropy. So simple point solvers cannot be applied. Therefore, following ideas from [Hac89, Hac9], in [Ste9, Ste95] we made a further direct sum decomposition of each V into (+) d d subspaces, using tensor products and again based on orthogonality with respect to discrete scalar products. Now the reduced boundary value problems on

4 ROB STEVENSON these smaller subspaces can be solved using point solvers yielding a robust preconditioner. A similar method and results were obtained by Griebel and Oswald in [GO95b]. Their preconditioner is based on the L -orthogonal decomposition which results into a somewhat less ecient algorithm since the required prewavelet bases have larger mass. In this paper we consider the question of robustness for anisotropic equations on linear nite element spaces in two dimensions. In this case, a decomposition of V as discussed above cannot be made since now the nite element space is not a tensor product space. Instead, we mae a direct sum decomposition of V into three subspaces again orthogonal with respect to the discrete scalar product. Although the reduced boundary value problems on these spaces are still not uniformly well-conditioned and thus point solvers cannot be applied, it will turn out that it is sucient to solve these problems with a line solver. At least for uniform triangulations, the resulting preconditioner will be robust with respect to anisotropy in the three directions of the grid lines. The remainder of this paper is organized as follows. In Section, we discuss general additive Schwarz preconditioners. The direct sum decomposition central in this paper together with various theoretical and practical aspects of the corresponding hierarchical basis preconditioner are treated in Section 3. In Section, further direct sum decompositions of the subspaces generating our decomposition are introduced yielding robust preconditioners for anisotropic equations. Numerical examples are given at the end of Sections 3 and. Following [u9], we shall use the notations <, > and =. When we write x < y ; x > y and x 3 = y 3 ; then there exist constants C, c, c 3 and C 3, that are independent of relevant parameters as the level or the value of a singular perturbation parameter, such that x C y ; x c y and c 3 x 3 y 3 C 3 x 3 :. Preliminaries.. Finite element spaces. Throughout this paper, we consider a sequence of nested linear or multi-linear nite dimensional nite element spaces M 0 M : : : M J H 0(); where is a domain in R d. For use in x3 we impose the following conditions (T) The diameter of a underlying nite element of M (a d-simplex or d-parallelogram) that was generated by renement in the transition from M to M, divided by the diameter of the enclosing nite element of M 0 is equivalent to. (T) Elements that were not rened in the transition from M to M are never rened further. (T3) The diameter of an element divided by the radius of the largest ball contained in is bounded, uniformly in all elements on all levels. Note that we do not mae an assumption whether the subdivision of into nite elements is conforming or non-conforming.

5 A ROBUST HIERARCHICAL BASIS PRECONDITIONER ON GENERAL MESHES 5.. Additive Schwarz preconditioners. Following presentations from [u9, Yse93, GO95a], in this subsection we discuss general additive Schwarz preconditioners. Suppose we have to solve the variational problem nd u M J such that (.) a(u; v) = f(v) (v M J ); where a is a scalar product on M J and f is a linear functional on M J. Let M J = P N(J) H some space decomposition and, for 0 N(J) <, let b be an auxiliary scalar product on H. These ingredients dene the preconditioner which we will introduce in connection to a defect computation, that is, as a defect correction step. In doing so, we avoid introducing an additional scalar product for transferring the variational problem (.) into an operator equation, whereas at the same time the discussion of the implementation becomes easier. Given some approximation u old of the solution u, this defect correction step is dened by (.) u new = u old + where for 0 N(J), u is the solution of (.3) u ; b (u ; w) = f(w) a(u old ; w) (w H ): The use of the scalar products b replacing a models the application of inexact solvers on the subspaces H. Dening operators T : M J! H by b (T u; w) = a(u; w) (w H ), we have u = T (u u old ) where u is the solution of (.), and thus u u new = (I T )(u u old ): One easily veries that the preconditioned operator P T is symmetric positive denite (SPD) with respect to a(; ). We conclude that the convergence speed of the Conjugate Gradient method applied to the preconditioned system is determined in a well-nown manner by the spectral condition number ( P T ). Following [GO95a], we dene a norm jjj jjj on M J by and introduce the values min = jjjujjj = inf u H ;u= P u b (u ; u ) a(u; u) inf 0=uM J jjjujjj ; a(u; u) max = sup and (fm 0=uM J jjjujjj J ; ag; It can be proved that (.) ( T ) = (fm J ; ag; fh ; b g) fh ; b g) = max min : (cf. [GO95a] and the references cited there), which reveals the direct relation between the eectiveness of the preconditioner and the stability of the space decomposition.

6 ROB STEVENSON In case of a direct sum space decomposition M J = L H as we will consider in this paper, there holds and so jjj T ujjj = b (T u; T u) = min a(p T u; P T u) a( P T u; u) a(u; T u) = a( max ; T u; u) where the bounds are attained. By substituting u = ( P T ) w, we conclude that indeed ( P T ) = max min, which gives an easy proof of (.) in this case. A direct consequence of (.) is that for obtaining a qualitative estimate of ( P T ) we may replace the scalar products a and b by equivalent scalar products, since by doing that, the right-hand side of (.) only changes by a constant factor. Particularly interesting for singularly perturbed boundary value problems is the following: Suppose we have a direct sum space decomposition, two scalar products a () and a () on M J, and on each space V two auxiliary scalar products b (). Let us denote the and b () characteristic values of the decomposition using the scalar products a (i) and b (i) min. Then for ; 0 with + > 0, there holds and V respectively by (i) max and (i) (.5) (fm J ; a () + a () g; fh ; b () + b () g) maxf() minf () max; () maxg min; () ming : on M J.3. Implementation. Let f i g be a basis of M J. We denote by U the vector representing u M J with respect to f i g. Analogously, by U we denote the representation of u H with respect to some basis f i g of H (0 N(J)). Let p be the representation of the trivial embedding H! M J with respect to above bases, that is, p U is the representation of u H M J with respect to f i g. The stiness matrices A = A J and ~A are dened by <AU; V >= a(u; v) (u; v M J ) and < ~ A U ; V >= a(u ; v ) (u ; v H ) respectively, where <; > denotes a Euclidean scalar product. Note that ~A = r Ap, where r is the matrix adjoint of p. With F being the vector such that <F; V >= f(v) (v M J ), we conclude that the representation of the variational problem (.) with respect to f i g is given by (.) Let B be the matrix satisfying AU = F: <B U ; V >= b (u ; v ) (u ; v H ): This relation is usually used to dene b by taing B some easily invertible approximation of ~A. Analogously to (.), we nd that the representation of (.3) with respect to f i g

7 is given by A ROBUST HIERARCHICAL BASIS PRECONDITIONER ON GENERAL MESHES where G is dened by B U = G ; <G ; V > = f(v ) a(u old ; v ) = <F; p V > <AU old ; p V > = <r (F AU old ); V > (v H ), or G = r (F AU old ). So we have U = B r (F AU old ). In order to correct U old, also the function represented by U should be represented with respect to f i g. We conclude that the implementation of (.) is given by U new = U old + p B r (F AU old ): In our applications, H will be a subspace of M. Because of the nesting M 0 M : : : M J, the embeddings M! M J can be implemented eciently in a recursive way using the nodal bases on all nite element spaces. To obtain an ecient implementation of the embedding H! M and thus of the embedding H! M J, it is important that the basis functions of H can be expressed as linear combinations of a small number of nodal basis functions from M. 3. A new hierarchical basis preconditioner 3.. The space decomposition. In this section, we replace the general scalar product a by the scalar product D(u; v) := Z ' d i u@ i vdx; where ' is a positive function that is constant on the interiors of the nite elements corresponding to M 0. This function ' has the tas to cover jumps of the coecients of the dierential operator across boundaries of the elements on the coarsest level. Let us denote the collection of these elements by 0. To mae our estimates independent of the jumps and the sizes of the 0 0, we consider the weighted L -scalar product (u; v) L = 0 0 diam( 0 'uvdx; ) Z 0 and dene norms L = (; ) L and j j H = D(; ). We start with collecting some well-nown tools for analyzing multi-level methods (cf. e.g. [BY93]). On M, we have (3.) juj H < u L (inverse inequality): Further, there exists a < such that for l (3.) D(u; v) < l juj H v L (u M l, v M ) A large ' or a small 0 0 are dierent faces of the same problem. If is (linearly) enlarged with a factor p, then a dierential i with respect to new coordinates is given i.

8 8 ROB STEVENSON (strengthened Cauchy-Schwarz inequality). The L -orthogonal decomposition M J = L J W is dened by W = M L M ( ) and W 0 = M 0. Let Q : H 0()! M be the L -orthogonal projection onto M ( 0) and Q := 0. Then Q Q = (I Q )Q maps onto W and we conclude that u = P J (Q Q )u is the unique splitting of u M J in terms of functions from the spaces W. Several authors have proved that (3.3) juj H = jq 0 uj H + J = (Q Q )u L (u M J ); that is, (fm J ; D(; )g; fw 0 ; D(; )g + P J = fw ; (; ) L g) < (cf. [Osw9, Zha9, u9, Osw9] for uniform renements, [Wan9, BY93] for non-constant ', and [DK9, BY93] for non-uniform renements assuming additionally to (T)-(T3) some standard conditions on the renement procedure that prevent neighboring elements to dier more than a constant factor in size). Note that (3.) and (3.3) imply that for and u W (3.) u L = juj H : Our hierarchical basis preconditioner is based on a space decomposition very similar to the L -orthogonal decomposition stated above. Let be the set of (free-) nodal points belonging to M. We equip M with a discrete scalar product (3.5) (u; v) M = x w ;x u(x)v(x) and norm u M = (; ) M, with weights w ;x such that (3.) M = L (uniformly in ). ' Using that diam( 0 is constant on each nite element underlying M ), an obvious approach for constructing (; ) M Z is to apply the quadrature formula exact on M j n ' diam( 0 ) udx = 'vol( ) diam( 0 ) i= u(x i ); where x ; : : : ; x n are the vertices of. By summing over the elements, we get a quadrature formula exact on M. Possible resulting weights belonging to slave nodes can then be distributed over the free nodes on that side, since by denition of a slave node the value of u M in that node is some (convex) linear combination of the values in the free nodes. The resulting (; ) M satises (3.) (even without (T3)). Now we dene V = M (;)M M ( ) and V 0 = M 0. Then the space decomposition that underlies our hierarchical basis preconditioner is given by M J = L J V : In contrast to the L -orthogonal decomposition M J = L W, in the next subsection we will see that for general, possibly locally rened meshes, stable and local bases of the spaces V can easily be constructed. Moreover, also in model cases for which such bases of the

9 A ROBUST HIERARCHICAL BASIS PRECONDITIONER ON GENERAL MESHES 9 spaces W are nown, the basis functions of V can be expressed as a linear combination of a considerably smaller number of nodal basis functions than the basis functions of W. As pointed out in x.3, this will result in a more ecient implementation. Let Y : M +! M be the (; ) M+ -orthogonal decomposition onto M ( 0) and Y := 0. For 0 J, on M J we dene projections (3.) Z ( = Z (J) ) = (I Y )Y Y J : It is easily veried that Z maps onto V, P J Z = I and so u = P J Z u is the unique splitting of u M J in terms of functions from the spaces V. We note here that the mappings Z are important for theory, but that they do not enter into the implementation of the preconditioner that is discussed in x.3 and also at the end of x3.. We impose the condition (3.8) Z L L < (uniformly in J). One might thin that the property (3.) already implies (3.8), but unfortunately this is generally not true. Until now, we were able to verify (3.8) in model cases discussed in the next subsection concerning quasi-uniform meshes and '. In fact, for these cases we prove the stronger result (3.9) u L = J Z u L (u M J ). Yet, our numerical experiments discussed in x3.5 indicate that (3.9) is also valid in local renement cases and in cases where ' has large jumps. Assuming (3.3) and (3.8), we can prove the following analogue of (3.3) for our space decomposition. Theorem 3.. There holds juj H = jz 0 uj H + J = Z u L (u M J ). Proof. (Based on [u9, Appendix].) To avoid treating the term jz 0 uj H every time separately, we start with showing that for and u V, (3.0) so that jz 0 uj H + P J = Z u L V, u L = juj H ; = P J jz uj H. Let. By (3.), we have that on u L = u M = (I Y )u M (I Q )u M = (I Q )u L < juj H : This result together with (3.) shows (3.0). Let u M J and put u l = (Q l Q l )u. Then u l M l and so for l < J, we have Z u l = 0. For l J, it follows from (3.), (3.8) and (3.) that jz u l j H < Z u l L < u l L < l ju l j H :

10 0 ROB STEVENSON Writing u = P J u l, we now get that J jz uj H = J J < < < l;m=0 J l;m=0 J l;m= minfl;mg D(Z u l ; Z u m ) = J l;m=0 lm ju l j H ju m j H minfl;mglm ju l j H ju m j H J ju l j < H juj H l=0 by (3.3). On the other hand, from (3.) and (3.0), we obtain juj H = J ;l=0 which completes the proof. D(Z u; Z l u) < J ;l=0 minfl;mg jlj jz uj H jz luj H < D(Z u l ; Z u m ) J jz uj H 3.. Bases. In this subsection, we select bases of the spaces M J and V. The union over of the bases of the spaces V will also be a basis of M J, which is in particular a hierarchical basis. For 0 J, we equip M with nodal basis f ;x : x g, scaled in such a way that it is orthonormal with respect to (; ) M. The space V 0 equals M 0, and we equip both spaces with the same basis. For J, we equip V with basis (3.) where (3.) Indeed, from ;x (z) = f ;x : x n g; ;x = ;x ( ;x; ;y) M ;y : y ( ;y ; ;y ) M ( ;x (x) if z = x 0 if x = z n, we see that (3.) is an independent set, whereas using ( ;y ; ;z ) M = 0 for y; z and y = z, one may chec that ;x V. Note that e.g. for linear nite elements the sum over y in (3.) has only (at most) two non-zero terms, that is, in that case ;x can be expressed as a linear combination of (at most) three nodal basis functions of M. See Figure for a two-dimensional example. Representations of elements of M or V ( > 0) with respect to above bases will be most naturally identied with grid functions on or n respectively. Proposition 3.. The basis of V dened in (3.,3.) ( > 0) is uniformly L -stable (Riesz-basis).

11 A ROBUST HIERARCHICAL BASIS PRECONDITIONER ON GENERAL MESHES A A AA A x A A AA A A @ fg = fg = n fg = fy : ( ;x ; ;y ) M = 0g Figure. y for which ( ;x ; ;y ) M = 0 Proof. The stability of the basis of V 0 follows from (3.). Now let. For an underlying nite element of M, let [ ] be the enclosing element of M. The conditions (T)-(T3) imply that (3.3) For y, dene ;y M by ;y (x) = ;y L > ;y L conclude using (3.) that for y, (3.) vol( ) vol( [ ]) > : ( if y = x 0 if y = x. From (3.3), we have (y ). Since ;y = ;y ;y M and ;y M = p w ;y (cf. (3.5)), we ( ;y ; ;y ) M = w ;y ;y M ;y M = ;y M ;y M = ;y L ;y L > : Let u = P y c ;y ;y be a function from M. From (3.) and by writing ;y = P x ( ;x ; ;y ) M ;x, we get (3.5) y jc ;y j = (u ; u ) M = (u ; u ) M = x j x n j y ( ;x ; ;y ) M c ;y j y ( ;x ; ;y ) M c ;y j : Finally, let v = P x n d ;x ;x be a function from V. From (3.) and the denition of ;x we have v L = v M = x n jd ;x j + y j x n ( ;x ; ;y ) M e ;x j ;

12 ROB STEVENSON where e ;x = d ;x ( ;y ; ;y )M. Clearly, there holds v L > Px n jd ;x j. When we equip coordinate spaces with Euclidean scalar products, the mapping from (e ;x ) x n to ( P x n ( ;x ; ;y ) M e ;x ) y is just the adjoint of the operator that was shown to be bounded in (3.5), and thus it is bounded as well. From this and (3.) we conclude that y j ( ;x ; ;y ) M e ;x j < je ;x j < x n x n and thus that v L < Px n jd ;x j. x n jd ;x j 3.3. The preconditioner. Suppose the scalar product a is a non-negative linear combination of scalar products a () and a (), a = a () + a () ; where a () = D and a() = (; ) L. Theorem 3. and (3.9) show that (fm J ; ag; fv ; ag) < uniformly in + > 0, ; 0 (and J) (use (.5)). [Clearly, if instead of (3.9) only (3.8) holds, then this result is still valid for = 0.] Let ~A be the stiness matrix of a on V, that is, < ~A U ; V >= a(u ; v ); where U (V ) is the representation of u (v ) with respect to the basis of V. From (3.0) and Proposition 3., for > 0 we have ~A = I + I; and thus ~A = ( ~A )I. Finally, for > 0 dening b (u ; v ) =< B U ; V >, with B satisfying B = ( A ~ )I, e.g. B = diag( A ~ ) (Jacobi iteration) or B = ( A ~ )I (Richardson iteration), we obtain b = a on V and so J (fm J ; ag; fv 0 ; ag + fv ; b g) < : We denote the representations of the trivial embeddings M! M and V! M with respect to nodal bases on M and M and the basis (3.) on V by p 0 and p respectively. The operator p 0 maps grid functions on to grid functions on whereas p maps grid functions on n to grid functions on. Apart from possible dierent scalings, the operator p 0 is the familiar multi-grid prolongation, whereas e.g. for linear nite elements p is given by a three-point stencil. We note that since the nodal basis on M is orthonormal with respect to (; ) M, the relation M L (;)M V = M is equivalent to =

13 A ROBUST HIERARCHICAL BASIS PRECONDITIONER ON GENERAL MESHES 3 Im p 0 Im p with respect to the Euclidean scalar product. For the representation p of the trivial embedding V! M J, we have 8 p >< 0 p {z 0 } p = >: (J) p p 0 p {z 0 } J if 0 < J if = 0 Denoting by A the stiness matrix of a on M, for > 0 there holds A = r 0 A p 0 ; ~A (= r A J p ) = r A p ( ~A 0 = A 0 ), where r 0 and r are matrix adjoints of p 0 and p respectively. Using these formulas, necessary stiness matrices can be computed eciently and we obtain the following implementation of our preconditioner: Algorithm 3.3. function HB(J,F ) begin if J = 0 then U = A 0 F ; else U = p B J r F + p 0 HB(J ; r 0 F ); return U; end The only dierence with the classical hierarchical basis preconditioner is the choice of p (and as a consequence that of r, A 0 and the B 's). For the classical hierarchical preconditioner, p is just the trivial injection from the grid n to. 3.. Examples. In this subsection, we give examples of nite element spaces with discrete scalar products for which we were able to prove u L = P J Z u L (u M J ) ((3.9)) implying Z L < L ((3.8)), which latter result was used as a condition for Theorem 3.. Example 3.. In this example, M H 0((0; ) d ) is the multi-linear nite element space based on the subdivision of [0; ] d into f( + [0; ] d )h : 0 i < h ; N d g, where h = (+). Taing ' and following the general construction of (; ) M explained P below formula (3.), we dene (u; v) M = h d x u(x)v(x), where = h N d \ (0; ) d is the set of nodal points. The choice of (; ) M determines the decomposition M J = L V as well as the mappings p 0 and p introduced at the end of the previous subsection. In Figure, p 0 and p are represented in stencil notation for d =. These stencils show the respec- representations of basis functions of M or V in terms of basis functions of M. For d =, let us denote M, V and (; ) M by M (), V () and (; ) M () tively. Then for the general case, we have M = N d j= M () :, ( N d j= u j ; N d j= v j ) M =

14 ROB STEVENSON s s s s - s s s 8 8 s n n s - R p 0 p Q dj= (u j ; v j ) @ fg = Figure. p 0 and p in case of bilinear nite elements on a regular grid., and one may chec that as a consequence V = M f0;::: ;g d nf0;::: ;g d V ; f s g = n where V = N d j= V () j (cf. also Figure for the case d = ). In [Ste9, x3.] and [Ste95, x] it was proved that the decomposition M J = L f0;::: ;Jg V d of M J into (J + ) d subspaces is L -stable, that is, (fm J ; (; ) L g; P f0;::: ;Jg dfv ; (; ) L g) <. The clustering lemma stated below says that stability of a direct sum decomposition cannot be lost by clustering several subspaces together. We conclude that (3.9) is valid. Lemma 3.5 is a special case of a result mentioned in [GO95a, x5]. Lemma 3.5. Let M J = L N(J) P M() l=0 H ;l be a decomposition of M J into P N(J) (M()+) subspaces and let a be a scalar product on M J. Then for the space decomposition obtained by for each clustering the spaces H ;0 ; : : : ; H ;M(), we have N (J) (fm J ; ag; = P M() f M() l=0 H ;l ; ag) (fm J ; ag; N(J) fh ;l ; ag) : Proof. Put H l=0 H ;l. From the fact that M J is a direct sum of the spaces H 0 ; : : : ; H N(J), it follows that for each the characteristic values () min, () max of the decomposition H = P l H ;l are in the interval [ min ; max ], where min, max are the characteristic values of the decomposition M J = L P l H ;l. The remainder of the proof follows easily from the denitions. Example 3.. Let 0 be a nite collection of triangles of type +f(x; y) [0; ] : y xg or + f(x; y) [0; ] : y xg where Z. Let R be the interior of [ 0, where we assume that is connected and contains at least one vertex of some 0. For > 0, we dene to be the collection of triangles that arises by breaing each into four congruent triangles. For 0, M H 0() is the linear nite element space corresponding to. Again taing ' and following the general construction of (; ) M proposed below P formula (3.), we dene (u; v) M = h x u(x)v(x), where = h Z \ with h = (+) is the set of nodal points. The resulting p 0 and p are represented in Figure 3. M() l=0

15 s - s s s s A ROBUST HIERARCHICAL BASIS PRECONDITIONER ON GENERAL MESHES 5 n s s n n s - R p 0 p R ns fg = Figure 3. p 0 and p in case of linear nite elements on a regular grid. f s g = n For comparison, we note that in [KO95] for this example stable bases of the subspaces of the L -orthogonal decomposition have been constructed consisting of basis functions that are linear combinations of ten nodal basis functions. Moreover, there it was argued that a number smaller than ten cannot be expected. For a proof of (3.9) in this case we refer to Appendix A. The same decomposition M J = L V and bases on the spaces V as in this example were used by Dahmen, Kleemann, Prossdorf and Schneider in [DKPS]. Their starting point was not the orthogonality of spaces with respect to discrete scalar products, but the so-called moment conditions; for \interior" x n, the basis functions ;x (cf. (3.)) appear to be L -orthogonal to polynomials of degree less or equal to one. As a consequence, with respect to the hierarchical basis of M J dened as the union of the bases of the V, the entries of the stiness matrices of certain integral and pseudo-dierential operators show a fast decay away from the diagonal (see [DKPS, DKPS9]). Therefore, these (dense) matrices can be well approximated by sparse matrices (matrix compression). Our stability result (3.9) seems to be new. Both Examples 3. and 3. deal with regular elements (hypercubes or equilateral, rightangled triangles). In both cases the employed discrete scalar products are equivalent to (; ) L ((3.)), and it appeared that the corresponding decomposition M J = L V is L - stable ((3.9)). Finally in this section, we show that both properties are maintained under two types of transformations of the elements. Let : R d! R d be an invertible mapping. Dene ~ = and, mapping functions on onto functions on ~, by u = u. Put ~M = M and ~ = which is the set of (free) nodal points belonging to ~M. For some discrete scalar product (; ) ~M as in (3.5), let ~V = ~M (;) ~M ~M ( ) and ~V 0 = ~M 0. First, we consider the case that is some linear (or ane) transformation. We tae (~u; ~v) ~M = jd j ~x ~ w ;(~x) ~u(~x)~v(~x) (~u; ~v ~M ): Then from (u ; v ) ~M = jd j (u; v) M and (u ; v ) L ( ) ~ = j jd (u; v) L (),

16 ROB STEVENSON we have ~M = L ( ~ ) uniformly in and, ~ V = V and (f ~M J ; (; ) L ( ~ )g; P f ~V ; (; ) L ( ~ )g) = (fm J ; (; ) L ()g; P fv ; (; ) L ()g): As a second case, we consider continuous that are C -dieomorphism on the interiors of () for each underlying element of M 0. We tae (~u; ~v) ~M = ~x ~ w ;(~x) ~u(~x)~v(~x) (~u; ~v ~M ): Note that this scalar product does not correspond to the application of quadrature formulas on () that are exact on ~M j ( ). Clearly, we have (u ; v ) ~M = (u; v) M and thus ~V = V. Furthermore, there holds and so jdj u L ( ~) u L () jd j ; (f ~M J ; (; ) L ( ~ )g; P f ~V ; (; ) L ( ~ )g) (jdj jd j ) (fm J ; (; ) L ()g; P fv ; (; ) L ()g): We conclude that the results (3.) and (3.9) for Examples 3. and 3. carry over to quasiuniform meshes Numerical results. We present numerical computed condition numbers of our additive Schwarz preconditioner for a local renement example and an example with a discontinuous coecient. Both examples concern linear nite elements on a two-dimensional domain. Example 3.. Let be the collection of two triangles f(x; y) [0; ] : y xg and f(x; y) [0; ] : y xg. For 0, we dene as the collection of triangles that arises by for ` breaing each triangle from into four congruent triangles and for > ` by breaing only triangles from contained in [0; ( )`] into four congruent ones. For 0, let M H0((0; ) ) be the linear nite element space corresponding to. We tae and dene the scalar products (; ) M according to the construction exposed below formula (3.) (see Figure ). The choice of (; ) M determines the decomposition M J = L V as well as the mappings p 0 and p. We numerically computed the characteristic values max, min and the condition number of the decomposition for the bilinear forms a(u; v) = R uv and a(u; v) = R P i u@ i v using auxiliary scalar products b on V dened by one step of Jacobi iteration. The results are given in Tables and. The rst column (J ` = 0) of both tables concerns the Although (3.) and (3.9) apparently do hold uniformly in jd j, this is not true for (3.), (3.3) and (3.) since these results rely on (T3). As a consequence, Theorem 3. cannot be applied. Remarably, the results of the forthcoming section can coarsely speaing be summarized by saying that nevertheless in case of Examples 3. and 3., the decomposition ~M J = L ~ V is j j H -stable uniformly for linear transformations that do not corrupt a \maximum angle condition" (which is weaer than (T3)).

17 A ROBUST HIERARCHICAL BASIS PRECONDITIONER ON GENERAL MESHES h 3h 3:5h h h fg = HJ ` ` HHH Figure. Weights w x; in (u; v) M = P x w x; u(x)v(x) in case of Example 3. with = and ` =. 0 3 \h J " min max min max min max min max :8 : : : :3 :9 : :3 5:0 :5 :3 5:0 3 5 :5 :3 5: : :3 5:3 :3 :3 5:3 :3 :3 5:3 3 :3 :3 5: : :3 5:5 : :3 5:5 : :3 5:5 : :3 5: : : 5: : : 5: : : 5: 8 : : 5: :0 : 5:9 :0 : 5:9 :0 : 5:9 5 Table. Condition numbers for Example 3. for a(u; v) = R uv with one Jacobi iteration as a subspace solver. HJ ` ` HHH \h J " min max min max min max min max :5 :8 :3 :8 :3 :9 3 :3 :9 3 : :9 : 3:0 : 3:0 : 3:0 3 : 3: 5 :0 3: :0 3: :0 3: :0 3: :0 3:3 :0 3:3 :0 3: :0 3:3 :0 3: :0 3: :0 3: Table. As Table, but now for a(u; v) = R P i u@ i v. uniform renement case from Example 3. and the bounded numbers for a(u; v) = R uv from Table illustrate the theoretical result proved in Appendix A. The other columns of this table show that in case of additional local renement levels the condition numbers are hardly larger. Table shows that the condition numbers for a(u; v) = R P i u@ i v are bounded as well illustrating Theorem 3.. From (.5) we conclude robustness of the additive

18 8 ROB STEVENSON ( ) h ( ) h P PPP Pq h BM B B B B 0 8 h ( ) h h fg = Figure 5. Weights w x; in (u; v) M = P x w x; u(x)v(x) in case of Example 3.8 with =. Schwarz preconditioner for a(u; v) = R P i u@ i v + R uv with ; 0, + > 0. Example 3.8. In this example we tae 0 8 on [ ; 3 ] and on [0; ]n[ ; 3 ]. To satisfy the condition that is constant on the interiors of the nite elements corresponding to M 0 we need a ner initial triangulation than in the previous example. We dene M as the nite element space M + from Example 3. assuming uniform renements. We dene (; ) M again following the construction below (3.) (see Figure 5). Numerical computed condition numbers for a(u; v) = R uv and a(u; v) = R P i u@ i v are given in Table 3. A comparison between Table 3 and the rst column of Tables and shows that the J \h J " min : : :39 :38 :3 :3 :0 :9 :8 :9 max : :3 :3 :3 :3 : :8 :9 3: 3: 5:0 5:5 5:8 : : {z } a(u; v) = R uv {z } a(u; v) = R P i u@ i v Table 3. Condition numbers for Example 3.8 with one Jacobi iteration as a subspace solver. condition numbers for this discontinuous are hardly larger than those for. We also computed condition numbers for the function dened by 0 8 on [ ; ] [ [ ; 3 ] and on [0; ] n([ ; ] [ [ ; 3 ] ). In this case the jump interface has a socalled cross-point. For a(u; v) = R uv we found condition numbers that are almost equal to those in the left-hand part of Table 3, but for a(u; v) = R P i u@ i v the condition numbers appeared to be growing as function of J. Yet, this result is not in conict with

19 A ROBUST HIERARCHICAL BASIS PRECONDITIONER ON GENERAL MESHES 9 Theorem 3., since for a jump interface with a cross-point estimate (3.3), which was used as a condition for Theorem 3., is not necessarily valid (cf. [Wan9]). Tests with the BP-preconditioner showed condition numbers that grew in a similar way.. Robustness for anisotropic problems In this section we study the question of robustness for anisotropic equations of our space decomposition M J = L V, in particular for the case of two-dimensional linear nite element spaces. The reduced boundary value problems on the subspaces appear to be not uniformly well-conditioned as function of the anisotropy and so anyhow simple point solvers cannot be applied. Therefore, we will apply the idea of constructing an approximate solver based on a space decomposition orthogonal with respect to discrete scalar products again by decomposing each V into a number of subspaces that are orthogonal with respect to (; ) M... Multi-linear nite element spaces. Mainly as an introduction for the case of linear nite element spaces treated in the next subsection, in this subsection we consider the multi-linear nite element spaces M H0((0; ) d ) and scalar products (; ) M from Example 3. and review results obtained in [Ste9, Ste95] for a further decomposition of the spaces V. As in Example 3., we use notations M (), V (), (), () ;x and () ;x to denote concepts corresponding to the one-dimensional case. To avoid maing exceptions for the case = 0 every time, we use additionally the notations () 0;x = () 0;x and () = ;. In [Ste9, Ste95] it was proved that the decomposition (.) M J = is robust for anisotropic forms Z (.) a(u; v) = c 0 that is, (.3) M f0;::: ;Jg d V ; where V := (0;) d uv + d j= c j Z (fm J ; ag; do j= (0;) ju@ j v; with j fv ; ag) < : V () j ; c j > 0, c j 0; Moreover, it was shown that if V is equipped with tensor product basis f O j () j ;x j : x d () j= j n () j g; then the corresponding stiness matrix is uniformly well-conditioned, which means that point solvers can be applied, or equivalently, that (.) (fm J ; ag; x fspan O j () j ;x j ; ag) < : The resulting preconditioner is a modication of the Frequency Decomposition Multi-Level Method introduced by Hacbusch ([Hac89, Hac9]). For a similar preconditioner based on

20 0 ROB STEVENSON L -orthogonal decompositions equivalent results were obtained by Griebel and Oswald in [GO95b]. Decompositions lie (.) and (.) are restricted to tensor product spaces and can therefore not be applied to linear nite element spaces. Still for the multi-linear case from Example 3. but now with d =, we consider therefore the decomposition (.5) M J = V 0 M J M of M J into (3J + ) subspaces, where U ;(;0) = V () O M (), U ;(0;) = M () M U ; = f(;0);(0;);(;)g O O V () and U ;(;) = V () V () : The spaces U ;(;0), U ;(0;) and U ;(;) are mutually orthogonal with respect to (; ) M and U ;. The decomposition (.5) arises from (.) by in particular there holds V = L (;)M for each > 0 clustering the spaces V ;0 ; : : : ; V ; yielding U ;(;0) and V 0; ; : : : ; V ; yielding U ;(0;) (see Figure ). From Lemma 3.5 and (.3) we conclude that the decompo- V 0;0 V ;0 V ;0 q q q q V J;0 V 0; V ; V ; V J; V 0; V ; q q q q V ; q q q q V J;J V 0;J V ;J V J;J V J;J Figure. Clustering process applied to M J = L f0;::: ;Jg V yielding the decompositions M J = V 0 L L J= Lf(;0);(0;);(;)g U ; (,, ) and M J = L J V ( ) sition (.5) is robust assuming exact subspace solvers. We equip the spaces U ; with tensor product bases using the bases on M () and V (). The representation of an element of U ;(;0), U ;(0;) or U ;(;) with respect to these bases is most naturally identied with a grid function on ( () n () ) (), () ( () ) n () or ( () n () ) respectively. The representations p of the trivial embeddings U ;! M are depicted in Figure.

21 A ROBUST HIERARCHICAL BASIS PRECONDITIONER ON GENERAL @R n- p p p 0 p @R fg = fg = ( () n() ) () fg = () (() n() ) fg = ( () n() ) Figure. Representations of the basis functions of U ; nodal basis functions of M in case of bilinear elements. in terms of the The stiness matrices corresponding to U ;(;0) and U ;(0;) appear not to be uniformly well-conditioned. Yet, we do obtain a robust preconditioner for problem (.), that is, possible anisotropy in one of the two directions of the grid lines, when the reduced boundary value problems on U ;(;0) and U ;(0;) are solved with y-line and x-line Jacobi respectively, whereas on U ;(;) point Jacobi can be applied. Indeed, note that these Jacobi solvers correspond to the application of additive Schwarz preconditioners with respect to the decompositions U ;(;0) = L x () n() () ;x N M (), U ;(0;) = L x () n() M () N () ;x and U ;(;) = L () N () x( () n() ) ;x ;x. The resulting space decomposition of M J can be obtained by applying clustering to the decomposition from (.) and so robustness appears from Lemma Linear nite element spaces. In this subsection we consider the linear nite element spaces M based on regular triangulations and scalar products (; ) M from Example 3.. Following ideas from x., for > 0 we mae a direct sum decomposition of V into three spaces denoted by U ;(;0), U ;(0;) and U ;(;). In Figure 8 we dene these spaces U ; V together with bases on these spaces by giving representations of the basis functions in terms of the basis functions of V. Note that the stencils dening U ;(;0) and U ;(0;) n n - U ;(;0) U ;(0;) U ;(;) - Figure 8. Representations of the basis functions of U ; in terms of the basis functions of V in case of linear nite elements (symbols analogue to Figure ). (As usual, possible contributions from grid points outside should be left out.) n cannot be obtained from each other by exchanging x- and y-coordinates. Yet, replacing

22 ROB STEVENSON by and vice versa in all three stencils would give equivalent results. By expressing the basis functions of V in terms of the basis functions of M (use Figure 3), we obtain the representations of the basis functions of U ; in terms of the basis functions of M depicted in Figure 9. The same stencils were also applied in [Kat9]. - n n - - p 0 p 0 p Figure 9. Representations of the basis functions of U ; in terms of the nodal basis functions of M (p ) in case of linear nite elements. The remainder of this subsection is mainly devoted to a proof of the following theorem. Theorem.. Consider the linear nite element spaces M with scalar products (; ) M from Example 3. but now assume that = R. Let a(u; v) = c 0 ZR uv + Z R < c " # + c " # + c 3 " n #! " u ; " v v where P c > 0, c 0, that is, possible anisotropy in one of the three directions of the grid lines, and let b ; be the auxiliary scalar product on U ; derived from a(; ) corresponding to one y-line ( ), (x + y)-line ( ) or x-line (-) Jacobi iteration for = (; 0), (0; ) or (; ) respectively. Then (fm J ; ag; fv 0 ; ag + J = f(;0);(0;);(;)g fu ; ; b ; g) < (J + ): A practical implementation of the preconditioner described in above theorem is given by the following algorithm, where r denotes the matrix adjoint of p. Algorithm.. function HB(J,F ) begin if J = 0 then U = A 0 F ; else U =p 0 (y-line blocdiag(r 0 A J p 0 )) r 0 F + p 0 ((x + y)-line blocdiag(r 0 A J p 0 )) r 0 F + p (x-line blocdiag(r A J p )) r F + p 0 HB(J ; r 0 F ); return U; end

23 A ROBUST HIERARCHICAL BASIS PRECONDITIONER ON GENERAL MESHES 3 Remars.3. The proof of Theorem. involves estimates of the characteristic values max and min of the decomposition (cf. x.). Here we note that the estimate of max that we are going to derive can be generalized to bounded domains by expanding \boundary perturbations" around a zero boundary value using a discrete Taylor formula. However, until now we were not able to give a suitable estimate for min in case of a bounded domain. Although Theorem. promises only sub-optimality of the preconditioner, our numerical experiments on a bounded domain reported in the next subsection show bounded condition numbers uniformly in J and the anisotropy. The rotated anisotropic form a(u; v) = Z < " # " # c + s ( u ( )sc s + u ; " v v where c = cos(), s = sin(), [ ; ) and [0; ], satises the conditions of Theorem. when [ ; 0] and p + sin(j j+ ), or equivalently, when the resulting stiness matrix on M is an L-matrix. On [ ; 0], the range of! p + sin(j j+ ) is equal to [0; 3 p ] with minima in 0, and. A simple transformation of coordinates shows that the rotated anisotropic equation discretized on M leads to the same system of equations as Laplace' equation discretized on the linear nite element space corresponding to the grid that is rotated over an angle and afterwards is horizontally enlarged with a factor p (cf. [Ste93]). Clearly this procedure results in a grid that does not satisfy the minimum angle condition (T3) uniformly in. The special angles f ; ; 0g correspond to those transformed grids that nevertheless do have maximum angles that are bounded below uniformly in (cf. ). A decomposition V = L f(;0);(0;);(;)g U ; according to Figure 8 can also be made in case of non-uniform or locally rened triangulations. In view of the preceding considerations, assuming approximate solvers on the spaces U ; that solve for the \strong couplings", we may hope for robustness for the described anisotropic equations, or equivalently, for isotropic equations discretized on grids that possibly have small angles but that do satisfy an maximum angle condition. To prove Theorem., it is sucient to show that for ~c := (c 0 ; c ; c ; c 3 ) = (; 0; 0; 0), (0; ; 0; 0), (0; 0; ; 0) and (0; 0; 0; ) the condition number of the decomposition is O(J + ). The stiness matrices on the spaces M, U ;(;0), U ;(0;) and U ;(;) are given in stencil notation in Table, the stencils referring to underlying grids h Z, h ((Z + ) Z), h (Z (Z + )) and h (Z + ) respectively. From this table we conclude that for all four choices of ~c, the auxiliary scalar products b ; on U ; dened in Theorem. are equivalent, or for ~c = (; 0; 0; 0) even equal to a, which means that for our analysis we may replace them by a. The proof of the next lemma follows directly from the form of the p (Figure 9) and that of p 0 (Figure 3).

24 ROB STEVENSON ~c M 3 U ;(;0) 3 U ;(0;) 3 U ;(;) 5 5 h i h i h 8 5 h 8 h i 5 h h 5 h 8 5 h h 8 5 h 5 h 8 5 (; 0; 0; 0) (massmatrix) (0; ; 0; 0) h (0; 0; ; 0) h (0; 0; 0; ) h Table. Stiness matrices for the bilinear form from Theorem. Lemma.. There holds 3 V = L (;)M f(;0);(0;);(;)g U ;. Further, for T : M! M dened by (T u)(x) = u(x + h ) ( Z ), we have T (0; ) U ;(;0), T ( ; ) U ;(0;), T ( ;0) U ;(;) (;)M M : (Appendix A), For the case ~c = (; 0; 0; 0), from (fm J ; (; ) L g; P fv ; (; ) L g) < V = L (;)M U ; (Lemma.) and M = L ((3.)) we immediately obtain that (fm J ; ag; fv 0 ; ag + P J = PfU ; ; ag) <. In the sequel, we tae ~c = (0; ; 0; 0), that is, a(u; v) = u@ v: The remaining two cases can be treated analogously. From Lemma. and the form of the stiness matrix on M, we nd that (.) U ;(;0) a(;) M : Analogously to (3.), one may prove that that there exists a < such that for l (.) a(u; v) < l a(u; u) v L (u M l, v M ): On both U ;(0;) and U ;(;) the stiness- and mass-matrices are well-conditioned and so on these spaces we have (.8) a(v; v) = v L : 3 In contrast to the bilinear case, in case of a bounded domain the spaces U ; are generally not mutually (; )M -orthogonal. Yet, by construction, also on bounded domains we have U ; V, and so U ; (;)M M.