Numerische Mathematik

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1 Numer. Math. (1996) 75: Numerische Mathematik c Springer-Verlag 1996 Electronic Edition A preconditioner for the h-p version of the finite element method in two dimensions Benqi Guo 1, and Weiming Cao 2,, 1 Department of Applied Mathematics, University of Manitoba, Winnipeg, Manitoba R3T 2N2, Canada 2 Department of Mathematics, Shanghai University of Science and Technology, Shanghai , P.R. China Received August 15, 1995 / Revised version received November 13, 1995 Summary. A preconditioner, based on a two-level mesh and a two-level orthogonalization, is proposed for the h-p version of the finite element method for two dimensional elliptic problems in polygonal domains. Its implementation is in parallel on the subdomain level for the linear or bilinear (nodal) modes, and in parallel on the element level for the high order (side and internal) modes. The condition number of the preconditioned linear system is of order max(1+ln H ip i i h ) 2, i H i is the diameter of the i-th subdomain, h i and p i are the diameter of elements and the maximum polynomial degree used in the subdomain. This result reduces to well-known results for the h-version (i.e. p i = 1) and the p-version (i.e. h i = H i ) as the special cases of the h-p version. Mathematics Subject Classification (1991): 65F10, 65N30, 65N55 1. Introduction For large-scale computations in parallel environments, iterative solvers can be much more efficient than direct solvers. The condition numbers of the linear systems generated from finite element discritizations may be very poor, particularly for the h-p version. Hence preconditioning is very important and essential for the efficiency and effectiveness of the iterative solvers. The iterative solution techniques for the linear systems generated from the h- version or p-version of the finite element method have been widely studied in the past decades. Among them are the most successful preconditioned conjugate gradient methods. The efficiency of these solution techniques relies largely on Partially supported by National Science & Engineering Research Council of Canada under grant OGP Partially supported by the Postdoctoral Fellowship Fund of University of Manitoba under account , during his visit to Department of Applied Mathematics, University of Manitoba page 59 of Numer. Math. (1996) 75: 59 77

2 60 B. Guo, W. Cao the condition number of the preconditioned system. We mention here only a few classical results in the two dimensional case. For the h-version, Bramble et al. [7] proposed a class of preconditioners based on a representation of the H 1/2 00 norm on the interfaces of subdomains, which generally results in a condition number of order O(1+ln H h )2, with H and h are the diameters of the subdomains and the elements, respectively. Dryja and Widlund [10] proposed a preconditioner for the reduced interface problem, which also results in the same condition number as above. These results are widely used and generalized. For more details, see Dryja and Widlund [10,11], Xu [19]. On the other hand, Babu ska et al. [2] proposed a preconditioner for the p-version of the finite element method, based on the partial orthogonalization procedure. The resulting condition number is of order O(1+lnp) 2 with p being the polynomial degree used. Mandel studied extensively the preconditioning of the p-version, with extensions and many applications to elasticity and three-dimensional problems, see Mandel [15,16]. The preconditioning for the h-p version has been considered only in recent years. Ainsworth [1] proposed a preconditioner for the h-p version on quasiuniform meshes. It is based on a similar technique as in Bramble et al. [7] to define a representation of the H 1/2 00 norm on the interfaces of subdomains. The resulting condition number is reported to be of order O(1+lnp) 2 (1+ln Hp h )2. This result reduces to that of the h-version, when p = 1. However, it gives an order of O(1+lnp) 4 in the case of p-version, when h = H. The lack of optimality stems from the non-equivalence of the H 1/2 00 -norm and its representation, which involves an additional factor O(1+lnp) 2. Very recently, Oden et al. [17] proposed a general precondition technique for the h-p version of the finite element method. There non-uniform distribution of polynomial degrees, quasi-uniform subdomains and quasi-uniform partition of each subdomain with irregular partition points are used. The condition numbers are shown to be of order O( H Hp Hp Hp h (1+ln h )(1+lnp)) or O( h (1+ln h )(1+lnp)), for two different classes of preconditioners. However, for constructing the preconditioners, in both papers the subdomains are required to be quasi-uniform. Therefore the h-p version with non-uniform meshes, particularly geometric meshes, cannot be covered in their analysis. It is well-known that the geometric mesh is an essential feature of the h-p version of the finite element method. It eventually leads to the exponential convergence for the problems on non-smooth domains in engineering practices (see Babu ska and Guo [5, 14]). Besides, the restriction of quasi-uniform mesh on the preconditioning of the h-p version is not practical in an adaptive process. In this paper, we design a preconditioner for the h-p version of the finite element method in two dimensions, with non-uniform mesh and non-uniform distribution of element degrees. It is also applicable to the h-p version associated with the geometric mesh. The condition number of the preconditioned linear system is proved to be of order max(1+ln H ip i i h ) 2, H i is the diameter of i the subdomain Ω i, h i and p i are the diameter of elements and the maximum polynomial degree used in Ω i. This bound coincides with the well-known results page 60 of Numer. Math. (1996) 75: 59 77

3 A preconditioner for the h-p version of the finite element method in two dimensions 61 for both of the h-version when p i = 1 and the p-version when h i = H i. Thus, it fully covers the h-version and the p-version as special cases. An outline of this paper is as follows. In Sect. 2, we introduce two-level mesh (non-uniform subdomains and quasi-uniform partition of each subdomain) for a model problem on a polygonal domain, and the finite element space for its h-p version approximation. Then in Sect. 3, a preconditioner based on the two-level mesh and two-level orthogonalization (elimination of internal shape functions in every element and elimination of linear or bilinear nodal shape functions on every subdomain) is proposed. Its implementation can be accomplished in parallel on the element level and on the subdomain level, respectively. It is proved in Sect. 4 that the condition number of the preconditioned linear system is of order max(1+ln H ip i i h ) 2, the maximum is taken over all subdomains. i The application of the preconditioner to the h-p version with the geometric mesh is discussed in Sect Two-level mesh and finite element space Let Ω be a polygonal domain in R 2. L 2 (Ω) and H 1 (Ω) are the usual Sobolev spaces. Given f L 2 (Ω), consider the following Poisson equation with a homogeneous Dirichlet condition: (2.1) { u = f in Ω, u =0 on Ω. Define a bilinear form a Ω (, ): H 1 (Ω) H 1 (Ω) R 1 as a Ω (u,v)= u vdω. Then the weak formulation of (2.1) is to find u H0 1 (Ω), such that (2.2) a Ω (u,v)=(f,v), v H0(Ω), 1 Ω (, ) stands for inner product in L 2 (Ω), and H 1 0 (Ω) ={u H1 (Ω) u= 0on Ω}. We now consider the approximation of (2.2) by the h-p version of the finite element method. First we describe the partitioning of the domain Ω. In order to control the condition number of the resulting linear systems and create an efficient parallel iterative solver, we decompose Ω into a two-level mesh: (1) Coarse Mesh. We divide Ω into a family of non-overlapping triangles or quadrilaterals Ω i,1 i N, i.e., Ω = N i=1 Ω i. We will hereafter refer to the Ω i s as subdomains, and use H i to denote the diameter of the subdomain Ω i. Generally speaking, the sub-problems (described below) related to one subdomain, or a collection of subdomains, can be assigned to one processor of a parallel computer. (2) Fine Mesh. Each subdomain Ω i is further partitioned into a fine quasiuniform mesh {ω, j =1,,N i }, i.e., Ω i = Ni j =1 ω. ω is either a triangle or page 61 of Numer. Math. (1996) 75: 59 77

4 62 B. Guo, W. Cao a quadrilateral. We will hereafter refer to the ω s as elements, and use h i to denote the characteristic diameter of the elements in Ω i. In the above partitions, we require that both coarse mesh and fine mesh are regular, i.e., the intersection of any two subdomains (resp. elements) can only be a common vertex, or a whole common edge, or an empty set; any interior angle θ of subdomains (resp. elements) satisfies 0 <θ 0 θ θ 1 <π; and the ratio of the longest edge over the shortest edge of any quadrilateral subdomain (resp. element) is bounded from above by a constant; etc. (see Ciarlet [9]). Note that unlike in [1] and [17], we do not require the coarse mesh to be quasiuniform in the above partitions of Ω. The quasi-uniformity is only required in each subdomain, instead of in the whole domain. We call this kind of partition locally quasi-uniform. A typical locally quasi-uniform mesh is shown in Fig. 1. This requirement is much less restrictive than that in [1] and [17], and can be satisfied by most of the finite element meshes used in practice. The preconditioner associated with these locally quasi-uniform meshes, which will be described in Sect. 3, is applicable to the h-p version associated with the geometric mesh as well. We will elaborate on the details in Sect. 5. Fig. 1. Locally quasi-uniform mesh Fig. 2. Subdomain and element To distinguish the coarse and fine meshes, in this paper we will refer to the vertices and edges of an element ω as nodes and sides, while still use vertices and edges to refer to those of a subdomain Ω i. We also use L to denote the number of sides of an element (or edges of a subdomain), i.e., L = 3 if the element (or subdomain) is triangular, or L = 4 if it is quadrilateral. Furthermore, for a particular subdomain, we will use Γ l, 1 l L, to denote its edges, and for an element, we will use γ (with appropriate subscripts) to denote its sides. Clearly, Γ l = Ml γ l,m, the γ l,m s are the sides of some elements in the subdomain. See Fig. 2. page 62 of Numer. Math. (1996) 75: 59 77

5 A preconditioner for the h-p version of the finite element method in two dimensions 63 Next we describe the approximation subspace for the h-p version of the finite element method for (2.2). Let T = {(ξ,η) 0 <ξ<1, 0 <η<1 ξ}and Q = {(ξ,η) 0 <ξ,η<1}be the reference elements. For any positive integer p, denote by Pp 1 (T ) the set of polynomials defined on T of total degree p, and by Pp 2 (Q) the set of polynomials defined on Q of separate degree p (with respect to each variable). For simplicity, we use ˆω to denote the reference element T or Q, and P p (ˆω) to denote either Pp 1 (T )orpp(q), 2 with the understanding that ˆω = T if the element is triangular or ˆω = Q if the element is quadrilateral. Now on the reference element ˆω, assume that we are given L + 1 positive integers p (l),0 l L. We define the following three sets of shape functions (or modes) on ˆω: 1). The set Ψ [N] (ˆω)ofnodal shape functions. It is composed of the linear or bilinear functions which have the value one at one node of ˆω, and zero at all others; 2). The set Ψ [Sl ] p (l) (ˆω), l =1,,L,ofside shape functions. If ˆγ l is a side of ˆω, then a side shape function associated with ˆγ l is zero on ˆω\ˆγ l. Ψ [Sl ] p (l) (ˆω) is composed of all side shape functions in P p (l)(ˆω) associated with ˆγ l. 3). The set Ψ [B] p (0) (ˆω)ofinternal shape functions. These shape functions are in P p (0)(ˆω) and vanish on ˆω. For simplicity, we also use Ψ [N] (ˆω), Ψ [Sl ] (ˆω), and Ψ [B] (ˆω) to denote the spaces p (l) p (0) spanned by the corresponding set of shape functions. Note that the above definition does not imply a unique set of shape functions. There are many different ways to create shape functions satisfying the above conditions. We refer to Babu ska et al. [3] for details regarding the selection of shape functions. We define the polynomial subspace on the reference element ˆω as follows Ψ(ˆω)=Ψ [N] (ˆω) L l=1 Ψ [Sl ] p (l) (ˆω) Ψ [B] p (0) (ˆω). Now for any element ω, let F be the linear (or bilinear) mapping from the reference element ˆω onto ω. p (0) and p (l) are the maximum degrees for the internal and side modes on ω. Then the the approximation subspace in ω is defined as Ψ(ω )=Ψ [N] (ω ) Ψ [S] (ω ) Ψ [B] (ω ), Ψ [N] (ω ) = {ˆv F 1 ˆv Ψ [N] (ˆω)}; Ψ [Sl ] (ω ) = {ˆv F 1 ˆv Ψ [Sl ] (ˆω)}; p (l) Ψ [S] (ω ) = L l=1 Ψ [Sl ] (ω ); Ψ [B] (ω ) = {ˆv F 1 ˆv Ψ [B] (ˆω)}. Here we do not require the side functions with different sides of ω to have the same maximum degree. However we require that p (0) page 63 of Numer. Math. (1996) 75: 59 77

6 64 B. Guo, W. Cao (2.3) p (0) max 1 l L p(l). We define the approximation subspace in Ω as Ψ(Ω) ={v v ω Ψ(ω ), ω Ω} H 1 0(Ω). Also for any subdomain Ω i, letting p i = max p (0), we define Ψ(Ω i ) as the set 1 j N i of restrictions in Ω i of functions in Ψ(Ω), i.e., Ψ(Ω i )={v Ωi v Ψ(Ω)}. The h-p version approximation of (2.2) is to find u Ψ(Ω), such that (2.4) a Ω (u,φ)=(f,φ), φ Ψ(Ω). 3. Preconditioner 3.1. Two-level orthogonalization and decomposition of function spaces To build the preconditioner, we need a suitable decomposition of the approximation subspace Ψ(Ω). It can be done by the decomposition of the subspace Ψ(Ω i ) in every subdomain. Thus in the following, we will consider only the spaces in Ω i. First, we introduce a subspace spanned by the linear functions in triangular Ω i, or by the bilinear functions in quadrilateral Ω i, as follows: Ψ [V] (Ω i )={v vis linear or bilinear in Ω i }. In the preconditioner defined below, this space is used to define a problem on the coarse mesh to provide effective overall communication of information. It is a necessary mechanism for the preconditioning in problems involving a large number of subdomains, see, e.g., Widlund [18]. Note that for a triangular subdomain, Ψ [V] (Ω i ) is contained in the linear finite element space associated with the fine mesh, which consists of the pull-backs of linear functions defined in the reference triangle. But for a quadrilateral subdomain, Ψ [V] (Ω i ) is generally not contained in the bilinear finite element space associated with the fine mesh. However, if we assume (3.1) min 0 l L p(l) 2, ω Ω i, then it is not difficult to see that Ψ [V] (Ω i ) Ψ(Ω i ). Furthermore (3.2) Ψ(Ω i )=Ψ [V] (Ω i ) Ψ [0] (Ω i ), Ψ [0] (Ω i )={v Ψ(Ω i ) v= 0 at the vertices of Ω i }. page 64 of Numer. Math. (1996) 75: 59 77

7 A preconditioner for the h-p version of the finite element method in two dimensions 65 Next, we describe the decomposition of Ψ [0] (Ω i ) by a two-level orthogonalization. (1) For every element ω,1 j N i, we make its side shape functions orthogonal to its internal shape functions, with respect to a ω (, ). In other words, we decompose Ψ(ω ) into a direct sum as follows: Ψ(ω )=Ψ [N] (ω ) Ψ [S] (ω ) Ψ [B] (ω ), Ψ [S] (ω )= L Ψ l=1 [Sl ] (ω ); Ψ [Sl ] (ω )={v Ψ [Sl ] (ω )+ Ψ [B] (ω ) a ω (v, φ) =0, φ Ψ [B] (ω ) }. The above process is usually called the partial orthogonalization in the p-version, see, e.g., Babu ska et al. [2]. It is easy to see that a function in Ψ [Sl ] (ω )is uniquely determined by its values on the side γ l of ω. Now we consider in the subdomain Ω i the space of side shape functions after the above orthogonalization. We define Ψ [S] (Ω i )={v v ω Ψ [S] (ω ), 1 j N i } H 1 (Ω i ). Assume that Ω i = Mi γ m, and that ω (m) is an element in Ω i with γ m as one of its sides. Then we can further decompose Ψ [S] (Ω i )as (3.3) Ψ [S] (Ω i )= Ψ [SI ] (Ω i ) Ψ [S Γ ] (Ω i ), Ψ [SI ] (Ω i )={v v Ψ [S] (Ω i ), and v Ωi =0}; Ψ [S Γ ] (Ω i )= Mi Ψ [Sγm ] (Ω i ); Ψ [Sγm ] (Ω i )={v v Ψ [S] (Ω i ), and v = 0 on all sides except γ m }. Clearly the function in Ψ [Sγm ] (Ω i ) has the support in only one element adjacent to the edge of the subdomain, and can be uniquely determined by its values on the side lying on the edge of the subdomain. (2) For the subdomain Ω i, we define the space of all the nodal functions on the elements ω Ω i (except those associated with the vertices of Ω i ) Ψ [N] (Ω i )={v v ω Ψ [N] (ω ), 1 j N i } Ψ [0] (Ω i ) and its subspace of all the nodal functions associated with the nodes inside Ω i, Ψ [NI ] (Ω i )={v v Ψ [N] (Ω i ), and v Ωi =0}. Now we define a subspace of Ψ [N] (Ω i ) orthogonal to Ψ [NI ] (Ω i ) with respect to a Ωi (, ). In other words, we decompose Ψ [N] (Ω i ) into Ψ [NI ] (Ω i ) and its orthogonal complement Ψ [N Γ ] (Ω i ), i.e., (3.4) Ψ [N] (Ω i )=Ψ [NI ] (Ω i ) Ψ [N Γ ] (Ω i ); page 65 of Numer. Math. (1996) 75: 59 77

8 66 B. Guo, W. Cao Ψ [N Γ ] (Ω i )={v Ψ [N] (Ω i ), and a Ωi (v, φ) =0, φ Ψ [NI ] (Ω i ) }. The above procedure is equivalent to computing the Schur complement of the linear element approximation on Ω i, see, e.g., Bjørstad and Widlund [6]. Clearly, the function in Ψ [N Γ ] (Ω i ) is uniquely determined by its values at the nodes on Ω i. Thus (3.5) Ψ [N Γ ] (Ω i )= L l=1 Ψ [N Γ l ] (Ω i ), Ψ [N Γ ] l (Ω i )={v Ψ [N Γ ] (Ω i ) v vanishes on the edges of Ω i except Γ l }. Once a set of basis functions has been chosen, e.g., the Lagrange bases, the above two-level orthogonalization is indeed a transformation of the basis functions. The spaces with tilde are spanned by the transformed bases, while the spaces without tilde are spanned by the original ones. In computations these orthogonalizations can be accomplished, equivalently, by suitable matrix block eliminations. Furthermore, step (1) can be implemented in parallel for all the elements, and step (2) in parallel for all the subdomains. Step (1) and (2) are also independent of each other, and thus can be treated in parallel. By (3.2) (3.5), we summarize the decomposition of Ψ(Ω i ) as follows: (3.6) Ψ(Ω i )=Ψ [V] (Ω i ) L Ψ l=1 [N Γ ] l (Ω i ) Mi Ψ [Sγ l ] l=1 (Ω i ) Ψ [I] (Ω i ), Ψ [I] (Ω i )= Ψ [NI ] (Ω i ) Ψ [SI ] (Ω i ) Ψ [B] (Ω i ). For any v Ψ(Ω i ), we have (3.7) v = v V + v N Γ + v Sγ + v I, v N Γ = L l=1 v N Γ l, v Sγ = Mi v Sγm, v I = v NI + v SI + v B, with each component belonging to the corresponding subspace in Ω i listed above. The stiffness matrix corresponding to the subdomain Ω i is of the form VV VN Γ VS γ VI N Γ N Γ N Γ S γ N Γ I symm. S γ S γ S γ I, II N Γ1 N Γ1 N Γ1 N Γ2 N Γ1 N ΓL N Γ N Γ = N Γ2 N Γ2 N Γ2 N ΓL, symm. N ΓL N ΓL page 66 of Numer. Math. (1996) 75: 59 77

9 A preconditioner for the h-p version of the finite element method in two dimensions 67 S γ S γ = S γ1 S γ1 S γ1 S γ2 S γ1 S γmi S γ2 S γ2 S γ2 S γmi symm. S γmi S γmi N I N I N I S I N I B II = S I S I 0. symm. BB, Remark 3.1. If all the elements ω are triangles or parallelograms, then the mappings F ij from the reference elements ˆω to ω are linear. In this case the block N I B in the above matrix II is identical 0, since the linear and bilinear functions are orthogonal to internal functions by Green s formula Preconditioner With the decomposition of spaces described above, we can define the preconditioner as N C Ω (u,v)= C Ωi (u,v), (3.8) C Ωi (u,v)=a Ωi (u V,v V )+ i=1 L M i a Ωi (u N Γ l,v N Γl)+ a Ωi (u Sγm,v Sγm )+a Ωi (u I,v I ). l=1 The stiffness matrix in Ω i associated with the preconditioner is then of the form VV \N Γ N Γ [S γ S γ 0, II \N Γ N Γ = N Γ1 N Γ1 0 N Γ2 N Γ2, [S γ S γ = 0 N ΓL N ΓL S γ1 S γ1 0 S γ2 S γ2 0 S ΓMi S ΓMi. page 67 of Numer. Math. (1996) 75: 59 77

10 68 B. Guo, W. Cao 3.3. Implementation In preconditioned iterations, a major task is to compute the matrix-vector products C 1 r, C is the stiffness matrix corresponding to the preconditioner, and the r s are the residuals. Clearly such a product can be obtained by solving the following problem: Find w Ψ(Ω) such that (3.9) C Ω (w, φ) =(r,φ), φ Ψ(Ω). It can be accomplished by the following steps: (1) Solution of w V. It is clear that this step is essentially identical to solving a problem defined on the coarse mesh with linear or bilinear approximation. (2) Solution of w Nl for each edge of subdomains. By the definition of C Ω (, ), we see that these problems are completely independent for different edges, and hence can be treated in parallel. (3) Solution of w Sγm for each side lying on subdomain edges. These problems are also completely independent for different sides. (4) Solution of w I. This step essentially involves the solution of a homogeneous Dirichlet problem for each subdomain. Using the matrix obtained from the partial orthogonalizations (see Sect. 3.1), we can get the solution in two steps: (a) get the solution on all interior nodes and sides, (b) then get the solution in all elements, which can also be computed in parallel. 4. Estimate of condition number First we recall some notations for the Sobolev spaces and their norms. Let S R 1 or R 2 be an open interval or polygon, and denote by L 2 (S ) and H 1 (S ) the usual Sobolev spaces with the norm L2 (S ), H1 (S ) and seminorm H 1 (S ). Also denote by L (S ) the space of essentially bounded functions in S with the norm L (S ). For convenience, we will usually use instead of H1 (S ) the following H 1 -norm, v H 1 (S ) =( v 2 H 1 (S ) + 1 (diam(s )) 2 v 2 L 2 (S ) )1/2, which is invariant under scaling. In addition, we introduce the norm over the quotient space H 1 (S )/R 1 as follows v Ḣ 1 (S ) = inf v + α H α R 1 1 (S ). By Theorem of [9] and a scaling argument, it is easy to see that (4.1) C 1 v H 1 (S ) v Ḣ1 (S ) C 2 v H1 (S ) and that the equivalence constants C 1 and C 2 are independent of the diameter of S. page 68 of Numer. Math. (1996) 75: 59 77

11 A preconditioner for the h-p version of the finite element method in two dimensions 69 If S R 2, we denote by H 1/2 ( S ) the space of the traces of functions in H 1 (S ). A semi-norm for this space is { v(x) v(x ) 2 } 1/2 v H 1/2 ( S ) = S S x x 2 dσ(x) dσ(x ), σ(x) and σ(x ) are the arc length distances of the points x and x to a reference point on S. Let I be a connected part of S with end points x 1 and x 2, and denote by H 1/2 00 (I ) the interpolation space which is halfway between H0 1(I ) and L2 (I ). An equivalent norm for this subspace is (see Grisvard [12]) { v(x) v(x ) 2 v 1/2 H = (I ) 00 I I x x 2 dσ(x) dσ(x ) } 1/2 + ( v(x) 2 I x x 1 + v(x) 2 x x 2 )dσ(x). Next we list several lemmas for the analysis of the preconditioner. Lemma 4.1. Let γ be a side of the element ω, and let p = max 0 l L p(l). Then for any v Ψ(ω ), v L (γ) c(1+lnp ) 1/2 v H 1 (ω ), c is independent of the diameter of the element. Proof. It follows from Theorem 6.2 of [2] and a scaling argument. Lemma 4.2. Let v Ψ(Ω i ), then (4.2) v L (Ω i ) c(1+ln H ip i ) 1/2 v H h 1 (Ω, i) i H i is the diameter of Ω i,h i is the characteristic diameter of the elements in Ω i,p i is the the maximum degree of polynomials used in the elements in Ω i, and c is independent of H i, h i and p i. Proof. We note that this result is given as Theorem 2 in [17]. To make our paper more self-contained we nevertheless outline the proof. We note that by using Lemma 2.2 of [8] and a scaling argument, (4.3) u L (Ω i ) c ln ɛ 1/2 u H 1 (Ω i ) +ɛ( u L (Ω i ) +H i u L (Ω i )), ɛ (0, 1). Using Markov s inequality and a scaling argument in every element in Ω i,we obtain with c 0 1 that u L (Ω i ) c 0p 2 i h i u L (Ω i ). By substituting the above inequality into (4.3) and choosing ɛ = we obtain (4.2). h i 3c 0p 2 i Hi ( 1/3), page 69 of Numer. Math. (1996) 75: 59 77

12 70 B. Guo, W. Cao Lemma 4.3. For any v Ψ(Ω i ), let v V be its linear or bilinear interpolation at the vertices Q l, 1 l L, of Ω i. Then (4.4) v N H 1 (Ω i ) c(1+ln H ip i ) 1/2 v h H 1 (Ω i). i Proof. Let α be an arbitrary real number. Then by a simple calculation and Lemma 4.2, v V H 1 (Ω i ) = v V α H 1 (Ω i ) c max 1 l L vv (Q l ) α c v α L (Ω i ) c(1+ln H ip i h i ) 1/2 v α H 1 (Ω i), which leads to the conclusion by using (4.1). Next we define the piecewise linear or bilinear interpolation operator Π h over Ω i. Let F be the linear or bilinear mapping from the reference element ˆω to the element ω. For any continuous function v in Ω i, we define Π h v ω =(ˆΠˆv) F 1, ω Ω i, ˆv = v F, and ˆΠ ˆv is the linear or bilinear interpolation of ˆv at the nodes of ˆω. Lemma 4.4. For any v Ψ(Ω i ), (4.5) Π h v H 1 (Ω i ) c(1+lnp i ) 1/2 v H1 (Ω i). Proof. Consider an element ω Ω i, and let α j be an arbitrary real number. Then by Lemma 4.1, we have, similarly as in the proof of the previous lemma, Π h v H 1 (ω ) = Π h v α j H 1 (ω ) c v α j L ( ω ) c(1+lnp i ) 1/2 v α j H 1 (ω ), which, together with (4.1), implies Π h v H 1 (ω ) c(1+lnp i ) 1/2 v H 1 (ω ). (4.5) follows from summing up the above inequalities over all ω in Ω i. Lemma 4.5. Let I R 1 be an interval and p be a positive integer. Then for any v P p (I ) which vanishes at the two end points of I, (4.6) v 1/2 H (I ) v H (I)+c(1+lnp) 1/2 v 1/2 L (I), 00 c is independent of the length of I. Proof. It follows easily from Theorem 6.6 of [2] and a scaling argument. page 70 of Numer. Math. (1996) 75: 59 77

13 A preconditioner for the h-p version of the finite element method in two dimensions 71 There is also a result similar to the above lemma for the h-version. Let I R 1 be an interval of length H, and let T h be a quasi-unform decomposition of I with characteristic element size h. We denote by Λ h (I ) the space of piecewise linear functions based on T h. Then we have Lemma 4.6. For any v Λ h (I ) which vanishes at the two end points of I, (4.7) v H 1/2 00 (I ) v H 1/2 (I)+c(1+ln H h )1/2 v L (I), c is independent of H and h. Proof. It follows from the proof of Lemma 3.2 of [18]. Finally, we recall two lemmas regarding the discrete harmonic extension of the h-version and the p-version. Lemma 4.7. (Lemma 5.1 of [6]) For any edge Γ l of a subdomain Ω i, and any piecewise linear polynomial ψ defined on Γ l vanishing at its two end points, there exists v Ψ [N] (Ω i ), which coincides with ψ on Γ l, vanishes on the other edges of Ω i, and which satisfies (4.8) v H 1 (Ω i ) c ψ H 1/2 00 (Γ l ), c is independent of H i and h i. Lemma 4.8. Assume that (2.3) holds for the element ω. Then for any side γ l of ω, and any polynomial ψ P p (l)(γ l ) vanishing at the two end points of γ l, there exists v Ψ(ω ), which coincides with ψ on γ l, vanishes on the other sides of ω, and which satisfies (4.9) v H 1 (ω ) c ψ H 1/2 00 (γ l ), c is independent of p and the diameter of ω. Proof. Note that the H 1 semi-norm and H 1/2 00 norm are essentially invariant under F, since ω is a regular element. Hence by (2.3), a direct application of Theorem 7.4 (if ω is triangular) or Theorem 7.5 (if ω is quadrilateral) of Babu ska et al. [2] leads to the conclusion. We now prove the main theorem of this paper. Theorem 4.1. Assume that (2.3) holds for all elements, and that (3.1) holds for all quadrilateral subdomains. Then there exist positive constants c 1 and c 2 independent of H i, h i and p i, such that for any u Ψ(Ω) (4.10) c 1 a Ω (u, u) C Ω (u, u) c 2 max (1+lnH ip i ) 2 a Ω (u,u). 1 i N h i page 71 of Numer. Math. (1996) 75: 59 77

14 72 B. Guo, W. Cao Proof. (1) We consider the left hand inequality in (4.10). For any u Ψ(Ω), we have on the subdomain Ω i that (see (3.7)) u = u V + L M i u N Γ l + u Sγm + u I. l=1 By the Cauchy-Schwarz inequality, ( a Ωi (u, u) (L +3) a Ωi (u V,u V )+ L a Ωi (u N Γ l, u N Γ l ) l=1 ) +a Ωi ( Mi u Sγm, Mi u Sγm )+a Ωi (u I,u I ). From the facts that the support of u Sγm is in one element, and that there are at most L different u Sγm sharing one element as their supports, we have M i M i M i a Ωi ( u Sγm, u Sγm ) L a Ωi (u Sγm, u Sγm ), which completes the proof of the left hand inequality in (4.10). (2) We turn to the right hand inequality in (4.10). We first consider a Ωi (u V, u V ). Note that u V is the linear or bilinear interpolation of u at the vertices of Ω i. Thus by Lemma 4.3 (4.11) a Ωi (u V, u V ) c(1+ln H ip i ) u 2 H h 1 (Ω. i) i Next, we consider a Ωi (u N Γ l, u N Γ l ) for each edge Γ l of Ω i. Since the nodal functions associated with the nodes lying on Γ l have been made orthogonal to those with nodes interior to Ω i (with respect to a Ωi (, ) ),wefind (4.12) a Ωi (u N Γ l, u N Γl ) aωi (u, u ) for any u Ψ [N] (Ω i ) which coincides with u N Γ l on Ω i. By Lemma 4.7, there exists such a piecewise linear or bilinear function, which we denote again by u, satisfying (4.13) a Ωi (u, u ) c u 2. H 1/2 (Γ 00 l ) Since u N Γ l coincides with Π h u u V at all the nodes on Γ l, and both of them are piecewise linear functions, it follows that u = u N Γ l = Π h u u V on Γ l.we have from Lemma 4.6 u H 1/2 00 (Γ l ) = Π h u u V H 1/2 00 (Γ l ) Π h u u V H i (4.14) H 1/2 (Γ l)+c(1+ln ) 1/2 Π h u u V L (Γ h l). i We have for an arbitrary real number α page 72 of Numer. Math. (1996) 75: 59 77

15 A preconditioner for the h-p version of the finite element method in two dimensions 73 Π h u u V L (Γ l ) Π h u α L (Γ l)+ u V α L (Γ l) 2 u α L (Γ l), and by Lemma 4.2, c(1+ln H ip i h ) 1/2 u α H 1 (Ω. i i) By (4.1), we further have that Π h u u V L (Γ l ) c(1+ln H ip i ) 1/2 u h H1 (Ω i). i Let ˆΩ be a domain with unit diameter, and let ˆΓ be part of its boundary. It is easy to show that for any continuous function v H 1 ( ˆΩ), vanishing at the endpoints of ˆΓ, ˆv H 1 ( ˆΩ) c ˆv H 1 ( ˆΩ). By the trace theorem, ˆv H 1/2 ( Γ ˆ ) c ˆv H 1 ( ˆΩ). Therefore by a scaling argument, Π h u u V H 1/2 (Γ l ) c Π h u u V H 1 (Ω i ) c( Π h u H 1 (Ω i ) + u V H 1 (Ω i )). Using Lemma 4.3 and Lemma 4.4, we have (4.15) Π h u u V H 1/2 (Γ l ) c(1+ln H ip i ) 1/2 u h H1 (Ω i). i Thus the combination of (4.12)-(4.15) leads to (4.16) a Ωi (u N Γ l, u N Γl ) c(1+ln H i h i )(1+ln H ip i h i ) u 2 H 1 (Ω i). Now we consider Mi a Ωi (u Sγm, u Sγm ). Note that u Sγm has the support in only one element ω (m) with γ m as one of its sides. Therefore a Ωi (u Sγm, u Sγm )=a ω(m) (u Sγm,u Sγm ). Since the side functions are made orthogonal to the internal ones in every element separately, a ω (m) (u Sγm, u Sγm ) a ω (m) (u, u ), for all functions in Ψ(ω (m) ) which are identical to u Sγm on ω (m). By Lemma 4.8 there exists such a function, we denote again by u, satisfying (4.17) a ω (m) (u, u ) c u 2. H 1/2 (γ 00 m ) By Lemma 4.5, we further have u 2 H 1/2 00 (γ m ) u 2 H 1/2 (γ m ) + c(1+lnp i) u 2 L (γ m ). page 73 of Numer. Math. (1996) 75: 59 77

16 74 B. Guo, W. Cao Note that u = u Sγm = u Π h u on γ m, and that u is a polynomial of degree p i on ω (m). We have for an arbitrary real number α u L (γ m ) = u Π h u L (γ m ) u α L (γ m)+ Π h u α L (γ m) 2 u α L (γ m), and by Lemma 4.1, c(1+lnp i ) 1/2 u α H 1 (ω (m)), which, together with (4.1), implies (4.18) u L (γ m ) c(1+lnp i ) 1/2 u H1 (ω (m)). Furthermore, (4.19) u H 1/2 (γ m ) c u Π h u H 1 (ω (m)) c( u H 1 (ω (m)) + Π h u H 1 (ω (m))). Similarly, as above, we have (4.20) Π h u H 1 (ω (m)) c(1+lnp i ) 1/2 u H 1 (ω (m)). Hence, combining (4.17)-(4.20), we obtain a Ωi (u Sγm, u Sγm ) c(1+lnp i ) 2 u 2 H 1 (ω, (m)) and (4.21) M i a Ωi (u Sγm, u Sγm ) c(1+lnp i ) 2 Mi c(1+lnp i ) 2 u 2 H 1 (Ω i). u 2 H 1 (ω (m)) Finally, since u I = u u V L l=1 u Γl Mi u γm, we have (4.22) a Ωi (u I, u I ) c(1+ln H ip i ) 2 u h H 1 (Ω i). i The right hand inequality follows readily from (4.11), (4.16), (4.21) and (4.22). Remark 4.1. Condition (3.1) is only used to guarantee the inclusion Ψ [V] (Ω i ) Ψ(Ω i ) (see Sect. 3.1). In special cases, e.g., when Ω i is a parallelogram, this inclusion holds without requiring (3.1). Remark 4.2. The condition number bound given by (4.10) covers both of the h-version and the p-version as special cases. For p i 1, it reduces to the bound of the h-version discussed in [10]. In this case the orthogonalization is carried out only for the linear/bilinear functions associated with the nodes on the edges of subdomains. On the other hand, if each element is a subdomain, then H i h i, and (4.10) reduces to the bound of the p-version discussed in [2]. In this case the orthogonalization is carried out only for the side functions in every element. page 74 of Numer. Math. (1996) 75: 59 77

17 A preconditioner for the h-p version of the finite element method in two dimensions 75 Remark 4.3. The condition number bound given by (4.10) for the h-p version seems to be the best we can expect, because it reduces completely to those of the h-version and the p-version, as indicated in the previous remark. This result is much better than those in [1] and [17]. Remark 4.4. The preconditioning technique used in this paper is a proper combination of those for the h-version and the p-version, e.g., two-level mesh and two-level orthogonalization. Hence existing iterative solvers for either the h- version or the p-version of the finite element method can be changed to build one for the h-p version. Remark 4.5. In practical computations, if p i is large, then the work of orthogonalizing the linear/bilinear (nodal) functions in the subdomains is much less than that of orthogonalizing the higher order (side) functions in the elements. 5. Preconditioning on a geometric mesh In this section, we will concentrate on the applications of the preconditioner defined in Sect. 3 to the h-p version with a geometric mesh, because of the importance of this type of mesh for the finite element solution of problems on nonsmooth domains. For simplicity, we take as an example the case Ω =(0,1) (0, 1). We assume that the solution of (2.2) has a corner singularity only at the origin O =(0,0), and belongs to the countably normed space B 2 β (Ω), 0 <β<1 (we refer to [4,13] for the regularity results in the space B 2 β (Ω)). Usually the geometric mesh Ωn σ contains layers Ω i,1 i n, according to the distance to the origin, and there are several elements (triangular or quadrilateral) Ω,1 j J(i), in the i-th layer such that with a mesh factor σ (0, 1) (5.1) h = diam(ω ) σ n i+1, 1 j J (i), 1 i n, and { d1,j =0, 1 j J(1), (5.2) d = dist(o,ω ) σ n i, 1 j J(i),1<i n. The numbers J (i) of elements in the i-th layer 1 i n are uniformly bounded by an integer J 0. For a complete description of geometric meshes see [5,14]. A typical geometric mesh Ωσ n is shown in Fig. 3 with n =4, Ω n σ ={Ω 1 i n, 1 j J(i)}, with J (1)=1, J(2)=4, J(3) = J (4)=8. We now group the elements in the same layer into subdomains Θ i,k as follows: Θ 4,1 = 4 j =1 Ω 4,j, Θ 4,2 = 8 j=5 Ω 4,j, Θ 3,1 = 4 j =1 Ω 3,j, Θ 3,2 = 8 j=5 Ω 3,j, Θ 2,1 = 2 j =1 Ω 2,j, Θ 2,2 = 4 j=3 Ω 2,j, Θ 1,1 = Ω 1,1 page 75 of Numer. Math. (1996) 75: 59 77

18 76 B. Guo, W. Cao Let H i,k and h i,k be the diameter of the subdomain Θ i,k and the characteristic diameter of the elements in Θ i,k. Then H i,k /h i,k J 0. Usually J 0 is a small integer. By applying Theorem 4.1, we have the following bound for the condition number of the h-p version with a geometric mesh: Fig. 3. Geometric mesh Theorem 5.1. Let Ω n σ be a geometric mesh on a polygonal domain Ω associated with a distribution of element degree P = {p, 1 j J (i), 1 i n}. If each subdomain of Ω contains only the elements in one layer of the geometric mesh, then the preconditioner C Ω defined in (3.8) satisfies (5.3) c 1 a Ω (u, u) C Ω (u, u) c 2 (1+lnp) 2 a Ω (u,u), p = max p,c 1 and c 2 are constants independent of p and n. Remark 5.1. If a uniform distribution p ij =[µn] or a linear distribution p ij =[µi] with a degree factor µ>0 is used, then we have (5.4) c 1 a Ω (u, u) C Ω (u, u) c 2 (1+lnn) 2 a Ω (u,u), with c 1 and c 2 independent of n but not of µ. Here [α] denotes the smallest integer α. Remark 5.2. If the distribution of the element degree is linear, then the term (1+ln H i,kµi h ) in (4.10) is quite small for the subdomains in the inner layers, i,k e.g., i =1,2, and very big for subdomains in the outer layers, e.g., i = n, n 1. Also the number of degree of freedoms in the subdomains in the inner layers page 76 of Numer. Math. (1996) 75: 59 77

19 A preconditioner for the h-p version of the finite element method in two dimensions 77 is much smaller than those in the subdomains in the outer layers. To increase the efficiency, we may combine the elements in different layers, e.g., the first and second layers, into one subdomain. It will make the numbers of degree of freedoms in each subdomain more balanced, while not affecting the bound of the condition number in (4.10). References 1. Ainsworth, M. (1993): A preconditioner based on domain decomposition of h-p finite element approximation on quasi-uniform meshes. Math. Comput. Sci. Technical Reports 16, University of Leicester 2. Babu ska, I., Craig, A., Mandel, J., Pitkäranta, J. (1991): Efficient preconditioning for the p version finite element method in two dimensions. SIAM J. Numer. Anal. 28, Babu ska, I.,Griebel, M., Pitkäranta, J. (1989): The problem of selecting the shape functions for a p-type finite element. J. Comput. Appl. Math. 28, Babu ska, I., Guo, B.Q. (1988): Regularity of the solutions of elliptic problem with piecewise analytic data. Part I: Boundary value problems for linear elliptic equation of second order. SIAM J. Math. Anal. 19, Babu ska, I., Guo, B.Q. (1988): The h-p version of the finite element method for domains with curved boundaries. SIAM J. Numer. Anal. 25, Bjørstad, P.E., Widlund, O.B. (1986): Iterative methods for the solution of elliptic problems on regions partitioned into substructures. SIAM J. Numer. Anal. 23, Bramble, J., Pasciak, J., Schatz, A. (1986): The construction of preconditioners for elliptic problems by substructuring. I. Math. of Comput. 175, Bramble, J., Xu, J. (1991): Some estimates for a weighted L 2 projection. Math. of Comput. 56, Ciarlet, P.G. (1978): The finite element method for elliptic problems. North-Holland, Amsterdam 10. Dryja, M., Widlund, O.B. (1989): Some domain decomposition algorithms for elliptic problems. In: Hayes, L., Kincaid, D. (eds.) Iterative methods for large scale linear systems, pp Academic Press, San Diego, CA 11. Dryja, M., Widlund, O.B. (1990): Towards a unified theory of domain decomposition algorithms for elliptic problems. In: Chan, T.F., Glowinski, R., Périaux, J., Widlund, O.B. (eds.) Proceedings of The Third International Symposium on Domain Decomposition Methods for Partial Differential Equations. SIAM Philadelphia, PA 12. Grisvard, P. (1985): Elliptic problems in nonsmooth domains. Pitman, Boston 13. Guo, B.Q., Babu ska, I. (1986): The h-p version of the finite element method, part 1: the basic approximation results. Comput. Mech. 1, Guo, B.Q., Babu ska, I. (1986): The h-p version of the finite element method, part 2: General results and applications. Comput. Mech. 1, Mandel, J. (1990): Iterative solvers by substructuring for the p-version finite element method. Comput. Methods Appl. Mech. Engrg. 80, Mandel, J. (1990): Two-level domain decomposition preconditioning for the p-version finite element method in three dimensions. Internat. J. Numer. Methods Engrg. 29, Oden, J.T., Patra, A., Feng, Y.-S. (1994): Parallel domain decomposition solvers for adaptive h-p finite element methods. TICAM Report Widlund, O.B. (1988): Iterative substructuring methods: Algorithms and theory for elliptic problems in the plane. In: Glowinski, R., Golub, G.H., Meurant, G.A., Périaux, J. (eds.) Proceedings of The First International Symposium on Domain Decomposition Methods for Partial Differential Equations. SIAM Philadelphia, PA 19. Xu, J. (1992): Iterative methods by space decomposition and subspace correction. SIAM Rev. 34, page 77 of Numer. Math. (1996) 75: 59 77