We propose a multi-level type operator that can be used in the framework of operator (or Caldéron) preconditioning to construct uniform preconditioners for negative order operators discretized by piecewise polynomials on a family of possibly locally refined partitions. The cost of applying this multi-level operator scales linearly in the number of mesh cells. Therefore, it provides a uniform preconditioner that can be applied in linear complexity when used within the preconditioning framework from our earlier work [Uniform preconditioners for problems of negative order, Math. Comp. 89 (2020), 645–674].
In this work, we construct a multi-level type preconditioner for operators of negative orders that can be applied in linear time and yields uniformly bounded condition numbers. The preconditioner will be constructed using the framework of “operator preconditioning” studied earlier in e.g. [12, 3, 6, 10]. The role of the “opposite order operator” will be fulfilled by a multi-level type operator, based on the work of Wu and Zheng in .
For some 𝑑-dimensional domain (or manifold) Ω, a measurable, closed, possibly empty and an , we consider the Sobolev spaces
In order to create such a preconditioner, we will use the framework described in our earlier work . Given , we constructed an auxiliary space with such that, for defined by and some suitable “opposite order” operator , a preconditioner of the form is found. The space is equipped with a basis that, modulo a scaling, is biorthogonal to the canonical basis for so that the representation of is an invertible diagonal matrix.
With being the space of continuous piecewise linears w.r.t. 𝒯, zero on 𝛾, the above preconditioning approach hinges on the availability of a uniformly boundedly invertible operator , which is generally the most demanding requirement. For example, if and , a viable option is to take as the discretized hypersingular operator. While this induces a uniform preconditioner, the application of cannot be evaluated in linear complexity.
In this work, we construct a suitable multi-level type operator that can be applied in linear complexity. For this construction, we require 𝕋 to be a family of conforming partitions created by newest vertex bisection [7, 13]. In the aforementioned setting of having an arbitrary , this multi-level operator induces a uniform preconditioner , i.e., is uniformly well-conditioned, where the cost of applying scales linearly in . We also show that the preconditioner extends to the more general manifold case, where Ω is a 𝑑-dimensional (piecewise) smooth Lipschitz manifold, and the trial space is the parametric lift of a space of piecewise or continuous piecewise polynomials.
Finally, we remark that common multi-level preconditioners based on overlapping subspace decompositions are known not to work well for operators of negative order. A solution is provided by resorting to direct sum multi-level subspace decompositions. Examples are given by wavelet preconditioners, or closely related, the preconditioners from , for the latter assuming quasi-uniform partitions.
For , an optimal multi-level preconditioner based on a non-overlapping subspace decomposition for operators defined on the boundary of a 2- or 3-dimensional Lipschitz polyhedron was recently introduced in .
In Section 2, we summarize the (operator) preconditioning framework from . In Section 3, we provide the multi-level type operator that can be used as the “opposite order” operator inside the preconditioner framework. In Section 4, we comment on how to generalize the results to the case of piecewise smooth manifolds. In Section 5, we conclude with numerical results.
In this work, by , we mean that 𝜆 can be bounded by a multiple of 𝜇, independently of parameters which 𝜆 and 𝜇 may depend on, with the sole exception of the space dimension 𝑑, or in the manifold case, on the parametrization of the manifold that is used to define the finite element spaces on it. Obviously, is defined as , and as and .
For normed linear spaces and , in this paper, for convenience over ℝ, will denote the space of bounded linear mappings endowed with the operator norm . The subset of invertible operators in with inverses in will be denoted as . The condition number of a is defined as .
For a reflexive Banach space and being coercive, i.e.,
Given a family of operators ( ), we will write ( ) uniformly in 𝑖, or simply “uniform”, when
Let be a family of conforming partitions of a domain into (open) uniformly shape regular𝑑-simplices, where we assume that 𝛾 is the (possibly empty) union of -faces of . For , such partitions automatically satisfy a uniform 𝐾-mesh property, and for , we impose this as an additional condition. The discussion of the manifold case is postponed to Section 4.
Recalling that is a family of piecewise or continuous piecewise polynomials of some fixed degree w.r.t. 𝒯, let uniformly in . A common setting is that ( ) for some . We are interested in finding optimal preconditioners for , i.e., uniformly in , whose application moreover requires arithmetic operations.
Recall the space
2.1 Construction of Optimal Preconditioners
For the moment, consider the lowest order case of being either the space of piecewise constants or continuous piecewise linears. In , a space was constructed with and
As a consequence of (2.1), defined by ( ) is in uniformly. We infer that, once we have constructed uniformly, then, by taking
The aforementioned space is a subspace of , where is a “bubble space” with , such that the projector on defined by and is “local” and uniformly bounded, and the canonical basis of “bubbles” for is, when normalized in , a uniformly Riesz basis for . Because of the latter, defined by
Given some “opposite order” operator , by taking
2.2 Implementation of
Recalling the aforementioned bases , and for , and , respectively, equipping with the nodal basis and equipping , , and with the dual bases , , and , respectively, the representation of is the stiffness matrix , and the representation of is the matrix . It is given by
The above preconditioning approach is summarized in the following theorem.
Theorem 2.1 ([10, Section 3])
Given a family uniformly in , then for as described in (2.3), the operator from (2.2) is a uniform preconditioner. Furthermore, if the matrix representation , cf. (2.5), can be applied in operations, then the matrix representation of the preconditioner , cf. (2.4), can be applied in operations.
Because in (2.3) is given as the sum of two operators that “act” on different subspaces of , the condition number of the preconditioned system depends on the relative scaling of both these operators which can be steered by selecting the parameter 𝛽. A suitable 𝛽 will be selected experimentally.
Alternatively, [11, Proposition 5.1] shows that a value of 𝛽 is reasonable if it is chosen such that the interval bounded by the coercivity and boundedness constants of is included in that interval corresponding to or vice versa. Also these coercivity and boundedness constants can be approximated experimentally or by making some theoretical estimates.
Constructions of , and , and resulting explicit formulas for matrices , , , are derived in . For ease of reading, we recall these formulas below for the case that is the space of piecewise constants. For the continuous piecewise linear case, we refer to [10, Section 4.2].
2.2.1 Piecewise Constant Trial Space
For , we define as the set of vertices of 𝒯, and as the set of vertices of 𝒯 that are not on 𝛾. For , we set its valence . For , and with denoting the set of its vertices, we set .
If one considers as the space of discontinuous piecewise constants, i.e.
2.3 Higher Order Case
For higher order discontinuous or continuous finite element spaces , suitable preconditioners can be built either from the current preconditioner for the lowest order case by application of a subspace correction method (most conveniently in the discontinuous case where, on each element, the space of polynomials of some fixed degree is split into the space of constants and its orthogonal complement), or by expanding by enlarging the bubble space . While referring to  for details, we recall that, with either option, the construction of an optimal preconditioner that can be applied in linear complexity hinges on the availability of an operator uniformly in , that can be applied in linear complexity.
3 An Operator of Multi-level Type
In this section, we will introduce an operator of multi-level type. The operator is based on a stable multi-level decomposition of given by Wu and Zheng . Usually, such a stable multi-level decomposition is used as a theoretical tool for proving optimality of an additive (or multiplicative) Schwarz type preconditioner for an operator in . In this work, however, we are going to use their results for the construction of the operator for which it is crucial that its application can be implemented in linear complexity.
3.1 Definition and Analysis of
For , let 𝕋 be the family of all conforming partitions of Ω into 𝑑-simplices that can be created by newest vertex bisection starting from some given conforming initial partition that satisfies a matching condition .
With and , for , let be the number of bisections needed to create 𝑇 from its ancestor , and for , let . Notice that . For , let denote the -orthogonal projector onto .
The case can be included by letting 𝕋 be the family of a partitions of Ω that can be constructed by bisections from such that the generations of any two neighboring subintervals in any differ by not more than one.
For , set , and define
With this hierarchy of partitions, we define an averaging quasi-interpolator by
Theorem 3.1 ([14, Lemma 3.7])
For the averaging quasi-interpolator from (3.1), and , it holds that
In , the inequality “ ” was proven for the case , and . The arguments, however, immediately extend to , and .
The proof of the other inequality “ ” follows from well-known arguments. For some , let for . Then by the reiteration theorem, and for , on .
Let be written as with . Then, for , , we have
The relevance of the multi-level decomposition from Theorem 3.1 by Wu and Zheng lies in the fact that can only differ from in any as well as in only two  of its neighbors in (the endpoints of the edge on which 𝜈 was inserted).
Proposition 3.2 ([14, Lemma 3.1])
With, for , , it holds that, for , ; see Figure 1.
The proof from  given for generalizes to . Indeed, the arguments that are used are based on the fact that the basis for that is dual to the nodal basis takes equal values in all but one nodal point. This is a consequence of the fact that the mass matrix of the nodal basis for , and so its inverse, is invariant under permutations of the barycentric coordinates, which holds true in any dimension.
As a consequence of Proposition 3.2, we have
3.2 Implementation of
Since the operator is a weighted local projection, it allows for a natural implementation by considering , the space of discontinuous piecewise linears w.r.t. 𝒯. Recall the nodal basis for , and equip with the element-wise nodal basis.
Denote for the representation of the embedding into . For , let be the representation of the -orthogonal projector of onto , and let . For , let be the representation of the averaging operator defined by
Then the representation of from (3.2) is given by
By representing 𝒯 as the leaves of a binary tree with roots being the simplices of , computing for the sequence amounts to computing, while traversing from the leaves to the root, for any parent and both its children the orthogonal projection of a piecewise linear function on the children to the space of linears on the parent. For , the matrix representation of the latter projection is given in Figure 2.
The application of can be computed in operations.
Because the number of nodes in a binary tree is less than 2 times the number of its leaves, for , the computation of the sequence takes operations. From Proposition 3.2, recall that any vector in vanishes at so that the number of its non-zero entries is bounded by . Knowing already and , computing any non-zero entry of requires operations. ∎
We conclude that the operator , with above matrix representation , satisfies the requirements of Theorem 2.1.
4 Manifold Case
Let Γ be a compact 𝑑-dimensional Lipschitz, piecewise smooth manifold in for some with or without boundary . For some closed measurable and , let
Let 𝕋 be a family of conforming partitions 𝒯 of Γ into “panels” such that, for , is a uniformly shape regular conforming partition of into 𝑑-simplices (that, for , satisfies a uniform 𝐾-mesh property). We assume that 𝛾 is a (possibly empty) union of “faces” of (i.e., sets of type , where 𝑒 is a -dimensional face of ).
As in the domain case, a space can be constructed with
For the case that Γ is not piecewise polytopal, a hidden problem is, however, that above construction of requires exact integration of lifted polynomials over the manifold. To circumvent this problem, in , we have relaxed the condition of -biorthogonality of and to biorthogonality w.r.t. to a mesh-dependent scalar product obtained from the -scalar product by replacing the Jacobian on the pull back of each panel by its mean. It was shown that the resulting preconditioner is still optimal and that the expression for its matrix representation (for the moment without the representation of ), that was recalled in Section 2.2.1 for the piecewise constant case, applies verbatim by only reading as the volume of the panel.
It remains to discuss the construction of an operator of multi-level type, where it is now assumed that 𝕋 is a family corresponding to newest vertex bisection. An exact copy of the construction of given in the domain case would require the application of the panel-wise -orthogonal projector , cf. (3.1), which generally poses a quadrature problem. Reconsidering the domain case, the proof of [14, Lemma 3.7] (which provides the proof of the inequality “ ” in our Theorem 3.1) builds on the fact that, for being a sequence of uniformly refined partitions, the decomposition , where , is stable, uniformly in 𝐿, w.r.t. the norm on . This stability holds also true when the orthogonal complements are taken w.r.t. a weighted -scalar product for any weight 𝑤 with .
This has the consequence that, for the construction of the multi-level operator in the manifold case, we may equip with scalar product
5 Numerical Experiments
Let be the two-dimensional manifold without boundary given as the boundary of the unit cube, , . We consider the trial space of discontinuous piecewise constants. We will evaluate preconditioning of the discretized single layer operator .
The role of the opposite order operator in from Section 2.1 will be fulfilled by the multi-level operator from (3.2). Equipping with the nodal basis , the matrix representation of the preconditioner from Section 2.1 reads as
Equipping and with “energy-norms” or , respectively, we calculated the (spectral) condition numbers , where is the spectral radius, using the Lanczos method.
As initial partition of Γ, we take a conforming partition consisting of 2 triangles per side, so 12 triangles in total, with an assignment of the newest vertices that satisfies the matching condition. We fixed , being the value for which, for a relative small uniform refinement 𝒯 of , we found .
5.1 Uniform Refinements
Here we let 𝕋 be the sequence of (conforming) uniform refinements, that is, is found by bisecting each triangle from into 2 subtriangles using newest vertex bisection.
|dofs||sec / dof|
Table 1 shows the condition numbers of the preconditioned system in this situation. The condition numbers are relatively small, and the timing results show that the implementation of the preconditioner is indeed linear.
5.2 Local Refinements
Here we take 𝕋 as a sequence of (conforming) locally refined partitions, where is constructed by applying newest vertex bisection to all triangles in that touch a corner of the cube.
Table 2 contains results for the preconditioned single layer operator discretized by piecewise constants . The preconditioned condition numbers are nicely bounded, and the timing results confirm that our implementation of the preconditioner is of linear complexity, also in the case of locally refined partitions.
|dofs||sec / dof|
Funding source: Nederlandse Organisatie voor Wetenschappelijk Onderzoek
Award Identifier / Grant number: 613.001.652
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