Multilevel preconditioners for embedded enriched partition of unity approximations

Abstract

In this paper we are concerned with the non-invasive embedding of enriched partition of unity approximations in classical finite element simulations and the efficient solution of the resulting linear systems. The employed embedding is based on the partition of unity approach introduced in Schweitzer and Ziegenhagel (Embedding enriched partition of unity approximations in finite element simulations. In: Griebel M, Schweitzer MA, editors. Meshfree methods for partial differential equations VIII. Lecture notes in science and engineering, Cham, Springer International Publishing; 195–204, 2017) which is applicable to any finite element implementation and thus allows for a stable enrichment of e.g. commercial finite element software to improve the quality of its approximation properties in a non-invasive fashion. The major remaining challenge is the efficient solution of the arising linear systems. To this end, we apply classical subspace correction techniques to design non-invasive efficient multilevel solvers by blending a non-invasive algebraic multigrid method (applied to the finite element components) with a (geometric) multilevel solver (Griebel and Schweitzer in SIAM J Sci Comput 24:377–409, 2002; Schweitzer in Numer Math 118:307–28, 2011) (applied to the enriched embedded components). We present first numerical results in two and three space dimensions which clearly show the (close to) optimal performance of the proposed solver.

Introduction

The direct generalization and extension of the classical finite element method (FEM) to allow for the use of arbitrary non-polynomial basis functions as in partition of unity (PU) based approaches like XFEM/GFEM [1,2,3,4,5] usually requires a fair amount of implementational work within the original finite element (FE) code. Thus, the timely evaluation of novel generalizations of the FEM in large-scale industrial applications, which in general rely on commercial software packages, is usually not feasible. This issue can, however, be overcome with the help of the embedding approach presented in [6]. It allows for the non-invasive stable embedding of an arbitrary approximation space $V_{\mathrm {ENR}}$ into a classical FE space $V_{\mathrm {HOST}}$ and thereby enables the easy evaluation of novel generalizations of the FEM empolying arbitrary approximation functions in an industrial context. In [6] it was demonstrated that the approach is free from artifacts and yields substantial improvements in terms of accuracy. Note that this approach is very different from classical global–local techniques where you consider an independent auxilliary local problem. We, however, blend two function spaces $V_{\mathrm {HOST}}$ and $V_{\mathrm {ENR}}$ to discretize the global problem directly with a single larger function space $V_{\mathrm {BND}}$ which comprises $V_{\mathrm {HOST}}$ and $V_{\mathrm {ENR}}$, compare also [7,8,9].

In this paper we are concerned with the construction of highly efficient solvers and preconditioners for the linear system arising from the discretization of the global problem with this blended function space $V_{\mathrm {BND}}$. To this end, we are concerned with the non-invasive construction of multilevel preconditioners based on subspace correction methods [10] which are also referred to as Schwarz methods e.g. in the domain decomposition context [11]. We construct the coarsening process for $V_{\mathrm {BND}}$ with the help of an available (geometric) multilevel structure of $V_{\mathrm {ENR}}$ and a multilevel decomposition of $V_{\mathrm {HOST}}$ obtained by an algebraic multigrid (AMG) method in a non-invasive fashion. The remainder of this paper is structured as follows: we first quickly introduce the mathematical foundation of our embedding approach, the partition of unity method (PUM), in “A partition of unity method for the embedding of arbitrary approximation spaces in finite element spaces” and summarize the actual embedding procedure. In “Subspace correction methods” we introduce efficient subspace corrections preconditioners for the linear system arising from the discretization of the global problem by our blended function space $V_{\mathrm {BND}}$. The results of our numerical experiments with these preconditioners are presented in “Numerical results” before we conclude with some remarks in “Concluding remarks”.

A partition of unity method for the embedding of arbitrary approximation spaces in finite element spaces

The PUM was introduced in [1, 12] as a generalization of the FEM and is based on [13]. The abstract ingredients which make up a PUM space

$$\begin{aligned} V^{\mathrm {PU}} := \sum _{i=1}^N \varphi _i V_i = {\text {span}} \langle \varphi _i \vartheta _i^m\rangle ; \end{aligned}$$

(1)

are a partition of unity (PU) $\{\varphi _i:i=1,\ldots ,N\}$ and a collection of local approximation spaces $V_i:=V_i(\omega _i) := {\text {span}} \langle \vartheta _i^m\rangle _{m=1}^{d_{V_i}}$ defined on the patches $\omega _i:={\text {supp}}(\varphi _i)$ for $i=1,\ldots ,N$. Thus, the shape functions of a PUM space are simply defined as the products of the PU functions $\varphi _i$ and the local approximation functions $\vartheta _i^m$. The PU functions provide the locality and global regularity of the product functions $\varphi _i \vartheta _i^m$ whereas the functions $\vartheta _i^m$ equip $V^{\mathrm {PU}}$ with its approximation power. Note that there are no constraints imposed on the choice of the local spaces $V_i$, i.e. they are completely independent of each other. Thus, this very local interpretation of the PUM approach allows to utilize local a priori information about the sought solution by using so-called enrichment functions or physics-based basis functions in general [5]. Here, we usually employ local approximation spaces $V_i$ of the form $V_i = {\mathcal {P}}_i + {\mathcal {E}}_i$ where ${\mathcal {P}}_i$ denotes a space of local polynomials and ${\mathcal {E}}_i$ accounts for non-smooth local features such as kinks, discontinuities and singularities of the solution on the patch $\omega _i$. In our setting, however, we will not follow this local approach but we take a more global point of view which is in spirit closer to a domain decomposition line of thought, see e.g. [11], and can be viewed as a generalization of [7,8,9], see [6] for details.

Let us consider a very simple cover of the domain $\Omega$ into just two overlapping patches (or subdomains) $\Omega _0$ and $\Omega _1$ with respective PU functions $\Phi _0$ and $\Phi _1$, i.e. $\Phi _0+\Phi _1\equiv 1$ on $\Omega \subseteq \Omega _0\cup \Omega _1$. Then, let us choose the approximation space $V_0$ on the patch $\Omega _0$ to be a classical FE space defined on a respective mesh $\Omega _{0,h}$ which discretizes $\Omega _0$. According to the general PUM approximation theory this choice of $V_0$ imposes absolutely no constraints on our choice of the approximation space $V_1$ on the other subdomain $\Omega _1$. For instance, we could choose another FE space on a non-matching mesh $\Omega _{1,H}$ and blend these non-matching spaces smoothly together via the PUM [7, 8]. Another admissible choice for $V_1$ is the use of a locally enriched PUM space [6]. Throughout this paper we focus on the latter case where we blend an enriched approximation space with a classical FE space. Thereby, we can equip any classical FE simulation with stable enrichment capabilities via the PUM in a non-invasive fashion. To this end, let us introduce some more descriptive notation to identify the various components employed in our overall scheme. We refer to the two patches or subdomains by $\Omega _{\mathrm {HOST}}$ and $\Omega _{\mathrm {ENR}}$, compare Fig. 1. Moreover, we denote the respective function spaces defined on these subdomains by $V_{\mathrm {HOST}}$ and $V_{\mathrm {ENR}}$ where $V_{\mathrm {HOST}}$ is a classical FE space and $V_{\mathrm {ENR}}$ is an enriched approximation space. Note that in this simplistic two subdomain case we can rewrite the PU functions as $\Phi _{\mathrm {HOST}}:=\Phi$ and $\Phi _{\mathrm {ENR}}:=(1-\Phi )$ with some non-negative weight function $\Phi$ with ${\text {supp}}(\Phi )=\bar{\Omega }_{\mathrm {HOST}}$.

Fig. 1

Sketch of smooth model problem in two dimensions depicting the employed partitioning of the domain $\Omega$ into $\Omega _{\mathrm {HOST, FT}}$ (drak gray), $\Omega _\Phi =\Omega _{\mathrm {HOST}}\cap \Omega _{\mathrm {ENR}}$ (light gray), and $\Omega _{\mathrm {ENR, FT}}$ (white). The boundary data for this model problem is given by $g=0$ on $\Gamma _D$ and $h=(1, 1)^T\cdot n$ on $\Gamma _N$ where n denotes the outer normal

With the help of this notation and (1) we define our blended global approximation space $V_{\mathrm {BND}}$ on the complete domain $\Omega$ by

$$\begin{aligned} V_{\mathrm {BND}} := \Phi _{\mathrm {HOST}} V_{\mathrm {HOST}} + \Phi _{\mathrm {ENR}} V_{\mathrm {ENR}} = \Phi V_{\mathrm {HOST}} + (1 - \Phi ) V_{\mathrm {ENR}} \end{aligned}$$

(2)

such that any element $v \in V_{\mathrm {BND}}$ can be written as

$$\begin{aligned} v_{\mathrm {BND}} = \Phi _{\mathrm {HOST}} v_{\mathrm {HOST}} + \Phi _{\mathrm {ENR}} v_{\mathrm {ENR}} \end{aligned}$$

with $v_{\mathrm {HOST}} \in V_{\mathrm {HOST}}$ and $v_{\mathrm {ENR}} \in V_{\mathrm {ENR}}$. Note that the general convergence theory of the PUM [1, 12] allows to obtain some straightforward error bounds for this blending approach. To this end, let us consider the following estimate from [14].

Theorem 1

Let $\Omega \subset \mathbb {R}^D$ be a Lipschitz domain. Let $\{\varphi _i:i=1,\ldots ,N\}$ be an admissible non-negative partition of unity defined on patches $\omega _i:={\text {supp}}(\varphi _i)$.

Let us further introduce the covering index $\lambda _{C_\Omega }:\Omega \rightarrow \mathbb {N}$ such that

$$\begin{aligned} \lambda _{C_\Omega } (x) := {\text {card}}(\{\omega _i \in C_{\Omega }:x \in \omega _i\}) \end{aligned}$$

(3)

and let us assume that

$$\begin{aligned} \lambda _{C_\Omega }(x) \le \Lambda \in \mathbb {N} \quad \text {for all } x\in \Omega \end{aligned}$$

(4)

with $\Lambda$ independent of the number of patches N. Let a collection of local approximation spaces $V_i = {\text {span}}\langle \vartheta _i^m \rangle \subset H^1(\omega _i)$ be given. Let $f \in H^1(\Omega )$ be the function to be approximated. Assume that the local approximation spaces $V_i$ have the following approximation properties: On each patch $\Omega \cap \omega _i$, the function f can be approximated by a function $v_i \in V_i$ such that

$$\begin{aligned} \Vert f-v_i\Vert _{L^2(\Omega \cap \omega _i)} \le \hat{\epsilon }_{i}, \quad \text {and} \quad \Vert \nabla (f-v_i)\Vert _{L^2(\Omega \cap \omega _i)} \le \tilde{\epsilon }_{i} \end{aligned}$$

(5)

hold for all $i=1,\ldots ,N$. Then the function

$$\begin{aligned} v := \sum _{i=1}^N \varphi _i v_i \in V^\mathrm{PU} \subset H^1(\Omega ) \end{aligned}$$

satisfies the global estimates

$$\begin{aligned} \displaystyle \Vert f-v\Vert _{L^2(\Omega )}\le & {} \displaystyle \Bigg ( \sum _{i=1}^N \Vert \varphi _i\Vert _{L^\infty (\mathbb {R}^d)} \hat{\epsilon }^2_{i}\Bigg )^{1/2} , \end{aligned}$$

(6)

$$\begin{aligned} \displaystyle \Vert \nabla (f-v)\Vert _{L^2(\Omega )}\le & {} \displaystyle \sqrt{2} \Bigg ( \sum _{i=1}^N \Lambda \big (\Vert \nabla \varphi _i\Vert _{L^\infty (\mathbb {R}^d)} \hat{\epsilon }_{i}\big )^2 + \Vert \varphi _i\Vert _{L^\infty (\mathbb {R}^d)} \tilde{\epsilon }^2_{i} \Bigg )^{1/2} \,. \end{aligned}$$

(7)

In our setting we only have two approximation spaces $V_{\mathrm {HOST}}$ and $V_{\mathrm {ENR}}$ and let us, for the ease of notation, assume that both spaces ($i=\mathrm {HOST}, \mathrm {ENR}$) admit error bounds (5) of the form

$$\begin{aligned} \Vert f-v_i\Vert _{L^2(\Omega \cap \omega _i)} \le C h_{\mathrm {HOST}}^{p+1}, \quad \text {and} \quad \Vert \nabla (f-v_i)\Vert _{L^2(\Omega \cap \omega _i)} \le C h_{\mathrm {HOST}}^{p}. \end{aligned}$$

(8)

Then it is sufficient to ensure that the estimates

$$\begin{aligned} \Vert \nabla \Phi \Vert _{L^\infty ({\mathbb {R}}^d)} \le \frac{C_\nabla }{h_{\mathrm {HOST}}}, \quad \Vert \Phi \Vert _{L^\infty (\mathbb {R}^d)} \le C_\infty \end{aligned}$$

are satisfied by our PU function $\Phi$ to attain optimal convergence with the blended function space. Thus, our blending by the PUM provides optimal convergence even for very small overlap regions $\Omega _{\mathrm {HOST}} \cap \Omega _{\mathrm {ENR}}$.

The Galerkin discretization of a partial differential equation (PDE) using this blended function space $V_{\mathrm {BND}}$ yields the linear system

$$\begin{aligned} K_{\mathrm {BND}} \tilde{u}_{\mathrm {BND}} = \hat{f}_{\mathrm {BND}}, \end{aligned}$$

(9)

where $\hat{f}_{\mathrm {BND}}$ denotes the load vector and $\tilde{u}_{\mathrm {BND}}$ the respective coefficient vector of the solution. If we assume that the PU function $\Phi$ satisfies the so-called flat-top property

$$\begin{aligned} \Phi |_{\Omega _{\mathrm {HOST, FT}}} = \Phi _{\mathrm {HOST}}|_{\Omega _{\mathrm {HOST, FT}}} \equiv 1 \end{aligned}$$

for some subset $\Omega _{\mathrm {HOST, FT}} \subset \Omega _{\mathrm {HOST}} = \Omega _{\mathrm {HOST, FT}} \cup \Omega _\Phi$ with $\Omega _\Phi :=\Omega _{\mathrm {HOST}}\setminus \Omega _{\mathrm {HOST, FT}}$. Obviously, the other PU function $(1-\Phi )=\Phi _{\mathrm {ENR}}$ then satisfies

$$\begin{aligned} (1-\Phi )|_{\Omega _{\mathrm {ENR, FT}}} = \Phi _{\mathrm {ENR}}|_{\Omega _{\mathrm {ENR, FT}}} \equiv 1 \end{aligned}$$

for some $\Omega _{\mathrm {ENR, FT}} \subset \Omega _{\mathrm {ENR}} = \Omega _{\mathrm {ENR, FT}} \cup \Omega _\Phi$ with $\Omega _{\mathrm {HOST, FT}}\cap \Omega _{\mathrm {ENR, FT}}=\emptyset$ and $\Omega _{\mathrm {HOST}}\cap \Omega _{\mathrm {ENR}}=\Omega _\Phi$. With the help of the flat-top property of the PU we can thus introduce the splitting

$$\begin{aligned} V_{\mathrm {BND}} := V_{\mathrm {HOST, FT}} + \Phi _{\mathrm {HOST}} V_{\mathrm {HOST, FT}}^\bot + \Phi _{\mathrm {ENR}} V_{\mathrm {ENR, FT}}^\bot + V_{\mathrm {ENR, FT}} \end{aligned}$$

(10)

of our global blended function space $V_{\mathrm {BND}}$ into four components where any $v \in V_{\mathrm {HOST, FT}}$ satisfies ${\text {supp}}(v) \subset \bar{\Omega }_{\mathrm {HOST, FT}}$ and $V_{\mathrm {HOST, FT}}^\bot$ denotes the complement of $V_{\mathrm {HOST, FT}}$ in $V_{\mathrm {HOST}}$. The respective block-partitioning of the global stiffness matrix then reads as

$$\begin{aligned} K_{\mathrm {BND}} = \left( \begin{array}{cccc} K_{\mathrm {HF,HF}} &{} K_{\mathrm {HF,HF^\bot }} &{} 0 &{} 0 \\ K_{\mathrm {HF^\bot ,HF}} &{} K_{\mathrm {HF^\bot ,HF^\bot }} &{} K_{\mathrm {HF^\bot ,EF^\bot }} &{} 0 \\ 0 &{} K_{\mathrm {EF^\bot ,HF^\bot }} &{} K_{\mathrm {EF^\bot ,EF^\bot }} &{} K_{\mathrm {EF^\bot ,EF}} \\ 0 &{} 0 &{} K_{\mathrm {EF,EF^\bot }} &{} K_{\mathrm {EF,EF}} \\ \end{array} \right) . \end{aligned}$$

(11)

The load vector $\hat{f}_{\mathrm {BND}}$ and coefficient vector $\tilde{u}_{\mathrm {BND}}$ in this block-form are given by

$$\begin{aligned} \hat{f}_{\mathrm {BND}} = \left( \begin{array}{c} \hat{f}_{\mathrm {HF}} \\ \hat{f}_{\mathrm {HF^\bot }} \\ \hat{f}_{\mathrm {EF^\bot }} \\ \hat{f}_{\mathrm {EF}} \end{array} \right) \quad \text {and} \quad \tilde{u}_{\mathrm {BND}} = \left( \begin{array}{c} \tilde{u}_{\mathrm {HF}} \\ \tilde{u}_{\mathrm {HF^\bot }} \\ \tilde{u}_{\mathrm {EF^\bot }} \\ \tilde{u}_{\mathrm {EF}} \end{array} \right) . \end{aligned}$$

Note that the sub-matrix $K_{\mathrm {HF,HF}}$ is the classical FE stiffness matrix on the sub-domain $\Omega _{\mathrm {HOST,FT}}$ and thus can be provided by any FE package whereas all other sub-matrices in (11) need to be computed by the embedding code. Hence, our approach is completely non-invasive to a (commercial) FE package with respect to the disjoint partitioning of the domain $\Omega$ into $\Omega _{\mathrm {HOST,FT}} \subset \Omega$, which is discretized by the (commerical) FE package, and $\Omega \setminus \Omega _{\mathrm {HOST,FT}}$, compare Fig. 1. However, we actually employ a FE discretization on $\Omega _{\mathrm {HOST}} = \Omega _{\mathrm {HOST,FT}} \cup \Omega _{\Phi }$ which is obtained by merging the FE discretization on $\Omega _{\mathrm {HOST,FT}}$ provided by the HOST code and the discretization on $\Omega _{\Phi }$ provided by our embedding code, see [6] for details. Obviously, the overall computational effort associated with the assembly of the linear system (9) thus scales with the size of the overlap region and thus a small overlap region is preferable from this point of view.

Considering the disjoint partitioning of the domain $\Omega$ into $\Omega _{\mathrm {HOST,FT}}$, $\Omega _{\Phi }$ and $\Omega _{\mathrm {ENR,FT}}$ we can introduce the block-partitioning

$$\begin{aligned} K_{\mathrm {BND}} = \left( \begin{array}{ccc} K_{\mathrm {HF,HF}} &{} K_{{\mathrm {HF},\Phi }} &{} 0 \\ K_{{\Phi ,\mathrm {HF}}} &{} K_{{\Phi ,\Phi }} &{} K_{{\Phi ,\mathrm {EF}}} \\ 0 &{} K_{{\mathrm {EF},\Phi }} &{} K_{\mathrm {EF,EF}} \\ \end{array} \right) , \end{aligned}$$

(12)

where

$$\begin{aligned} K_{{\Phi ,\Phi }} := \left( \begin{array}{cc} K_{\mathrm {HF^\bot ,HF^\bot }} &{} K_{\mathrm {HF^\bot ,EF^\bot }} \\ K_{\mathrm {EF^\bot ,HF^\bot }} &{} K_{\mathrm {EF^\bot ,EF^\bot }} \\ \end{array} \right) , \end{aligned}$$

which may serve as a natural starting point for the development of classical (single level) domain decomposition solvers and preconditioners. We, however, are interested in the construction of multigrid-like solvers for (9) via the blending of respective multilevel sequences of $V_{\mathrm {HOST}}$ and $V_{\mathrm {ENR}}$ and thus employ the even more compact block-partitioning

$$\begin{aligned} K_{\mathrm {BND}} = \left( \begin{array}{cc} K_{\mathrm {H,H}} &{} K_{\mathrm {H,E}} \\ K_{\mathrm {E,H}} &{} K_{\mathrm {E,E}} \end{array} \right) , \quad \hat{f}_{\mathrm {BND}} = \left( \begin{array}{c} \hat{f}_{\mathrm {H}} \\ \hat{f}_{\mathrm {E}} \end{array} \right) , \quad \tilde{u}_{\mathrm {BND}} = \left( \begin{array}{c} \tilde{u}_{\mathrm {H}} \\ \tilde{u}_{\mathrm {E}} \end{array} \right) , \end{aligned}$$

(13)

where

$$\begin{aligned} K_{\mathrm {H,H}} := \left( \begin{array}{cc} K_{\mathrm {HF,HF}} &{} K_{\mathrm {HF,HF^\bot }} \\ K_{\mathrm {HF^\bot ,HF}} &{} K_{\mathrm {HF^\bot ,HF^\bot }} \\ \end{array} \right) . \end{aligned}$$

Note, however, that this disjoint partitioning of the matrix corresponds to an overlapping partitioning of the domain $\Omega$ into $\Omega _{\mathrm {HOST}} = \Omega _{\mathrm {HOST,FT}} \cup \Omega _{\Phi }$ and $\Omega _{\mathrm {ENR}} = \Omega _{\mathrm {ENR,FT}} \cup \Omega _{\Phi }$. To introduce our proposed multilevel solver for (9) based on the partitioning (13) let us shortly review the general components of such solvers in the context of subspace correction methods.

Subspace correction methods

The computational effort associated with the solution of linear systems like (9) account for a very large part (often even the largest) of the overall computational cost in any implicit or stationary simulation. Thus, the development of efficient linear solvers is of great practical relevance and is still an active research field today. Even though classical general purpose numerical linear algebra techniques such as (sparse) matrix factorizations, see e.g. [15], are widely used in practice, it is well-known that their computational complexity is not optimal and that specialized iterative linear solvers are needed to tackle large scale problems with millions of unknowns efficiently.

A very sophisticated class of iterative methods which not only show an optimal scaling in the storage demand but also in the operation count are so-called multilevel iterative solvers or (geometric) multigrid methods which are particular instances of subspace correction methods [10]. Note, however, that these multilevel and multigrid solvers are not general algebraic methods but involve a substantial amount of information about the discretization and possibly the PDE. Thus, the introduction of such a (geometric) multilevel solver in a commercial software package is very much invasive and typically infeasible. However, there exist extensions of (geometric) multigrid methods, so-called algebraic multigrid methods (AMG) [16,17,18,19], which can be used as a non-invasive plugin solver also in commercial software [20, 21]. Such AMG solvers are successfully utilized in many different application fields yet they are essentially designed for classical mesh-based piecewise linear discretizations and thus are in general not directly applicable to discretizations with arbitrary approximation functions, i.e. $V_{\mathrm {BND}}$ and $V_{\mathrm {ENR}}$. Therefore, no optimal linear solver for (9) is readily available and we need to take the specific construction of our blended approximation space $V_{\mathrm {BND}}$ into account when designing a respective iterative linear solver. To this end, we employ classical subspace correction techniques which can utilize splittings such as (2) and (10).

There are two main variants of subspace correction approaches: the parallel subspace correction (PSC) scheme (or additive Schwarz method) and the successive subspace correction (SSC) scheme (or multiplicative Schwarz method). Assuming a splitting

$$\begin{aligned} V = \sum _{i=0}^J V_i \end{aligned}$$

(14)

of a global function space V, the PSC iteration reads

$$\begin{aligned} \tilde{u} \leftarrow \tilde{u} + \sum _{i=0}^J B_i (\hat{f} - K\tilde{u}) \end{aligned}$$

(15)

with

$$\begin{aligned} B_i := P_i K_i^{-1} P_i^T, \quad \text {the prolongation} \quad P_i:V_i \rightarrow V, \end{aligned}$$

(16)

and $K_i$ denoting the stiffness matrix with respect to subspace $V_i$. The SSC scheme is defined by

$$\begin{aligned} \text {For } i=0,\ldots ,J:\quad \tilde{u} \leftarrow \tilde{u} + B_i (\hat{f} - K\tilde{u}) \end{aligned}$$

(17)

and thus involves a successive update of the residual $\hat{f} - K\tilde{u}$ after each subspace correction. Note that the use of the exact inverse $K_i^{-1}$ in (16) is not necessary. In fact, the use of approximate subspace solvers is usually advisable and much more common, i.e. we define

$$\begin{aligned} B_i := P_i W_i P_i^T \quad \text {with} \quad W_i \approx K_i^{-1}. \end{aligned}$$

(18)

The main ingredients which control the performance of the iterations (15) and (17) are the specific choices of the subspace splitting (14), the prolongations (16) and the approximate subspace solvers $W_i$ in (18) where it is important to note that we do not assume that (14) allows for a unique decomposition $v=\sum _{i=0}^J v_i$ of a function $v \in V$. In fact, the redundancy in the splitting (14) has a substantial impact on the convergence properties of (15) and (17).

In classical multigrid terminology the approximate subspace solvers $W_i$ in (18) are referred to as smoothers and the subspaces $V_i$ correspond to the employed approximation spaces defined on different refinement levels of the underlying mesh, i.e.  $V_J=V$ denotes the finest discretization space and $V_i$ with $i<J$ are referred to as coarse spaces. The role of the smoothers $W_i$ is to reduce high frequency error components whereas the corrections $B_i (\hat{f} - K\tilde{u})$ obtained on the coarser levels should reduce low frequency errors so that all error frequencies are efficiently reduced in each iteration.

In the following we first focus on the construction of the coarse spaces $V_i$ with $i<J$ for our finest discretization space $V_J=V=V_{\mathrm {BND}}$ and the definition of the respective prolongations $P_i:V_i \rightarrow V=V_J=V_{\mathrm {BND}}$. To this end, let us review an essential property that the prolongations $P_i$ and coarse spaces $V_i$ should satisfy. Since the role of the coarse spaces $V_i$ is the resolution of low frequency errors they should at least contain the constant functions, i.e. $1 \in V_i$, and the prolongations should be exact for the constant functions. Thus, we need to specify a coarsening process for the blended space $V_{\mathrm {BND}}=V_J$ such that the resulting coarse space $V_{J-1}$ contains constants and a respective prolongation $P_{J-1}:V_{J-1}\rightarrow V_{J}=V_{\mathrm {BND}}$ which preserves constants. To this end, let us consider the representation of the constant functions in $V_J=V_{\mathrm {BND}}$

$$\begin{aligned} 1_{\mathrm {BND}} = \Phi _{\mathrm {HOST}} \cdot 1_{\mathrm {HOST}} + \Phi _{\mathrm {ENR}} \cdot 1_{\mathrm {ENR}}, \end{aligned}$$

where $1_{\mathrm {HOST}} \in V_{\mathrm {HOST}}$ and $1_{\mathrm {ENR}} \in V_{\mathrm {ENR}}$ denote the constant function on the overlapping sub-domains $\Omega _{\mathrm {HOST}}$ and $\Omega _{\mathrm {ENR}}$ respectively. Therefore, we can localize the coarsening process for the space $V_{\mathrm {BND}}$ to the two spaces $V_{\mathrm {HOST}}$ and $V_{\mathrm {ENR}}$ and then define the prolongation as

$$\begin{aligned} P_{J-1} := \left( \begin{array}{cc} P_{\mathrm {HOST}, J-1} &{} 0 \\ 0 &{} P_{\mathrm {ENR}, J-1} \\ \end{array} \right) \end{aligned}$$

provided that the two prolongations $P_{\mathrm {HOST}, J-1}:V_{\mathrm {HOST},J-1} \rightarrow V_{\mathrm {HOST}}$ and $P_{\mathrm {ENR}, J-1}:V_{\mathrm {ENR},J-1} \rightarrow V_{\mathrm {ENR}}$ preserve constants in $V_{\mathrm {HOST}}$ and $V_{\mathrm {ENR}}$ respectively. Hence, let us now focus on the definition of a non-invasive coarsening process for $V_{\mathrm {HOST}}$ which comprises the FE space $V_{\mathrm {HOST,FT}}$ (handled by the HOST code) and the FE space $V_{\mathrm {HOST,FT}}^\bot$ via AMG techniques.

In general, AMG constructs a suitable coarse space $V_{J-1} \subset V_J$ and a respective prolongation $P_{J-1}:V_{J-1} \rightarrow V_J$ that preserves constants from the stiffness matrix $K_{J}$ obtained by the Galerkin discretization of the PDE using the space $V_J$. In our setting, however, the block $K_{H,H}$ of the stiffness matrix (13) does not correspond to the Galerkin discretization by $V_{\mathrm {HOST}}$ but by the space $\Phi _{\mathrm {HOST}}V_{\mathrm {HOST}}$ (which does not contain the constant function). Thus, only the block $K_{\mathrm {HF,HF}}$ of the matrix

$$\begin{aligned} K_{\mathrm {H,H}} := \left( \begin{array}{cc} K_{\mathrm {HF,HF}} &{} K_{\mathrm {HF,HF^\bot }} \\ K_{\mathrm {HF^\bot ,HF}} &{} K_{\mathrm {HF^\bot ,HF^\bot }} \\ \end{array} \right) \end{aligned}$$

is given by a classical FE discretization and all other blocks are computed using basis functions that are products $\Phi _{\mathrm {HOST}} \phi _k$ of our PU function $\Phi _{\mathrm {HOST}}$ and classical FE basis functions $\phi _k$. Applying AMG directly to $K_{\mathrm {H,H}}$ therefore does not provide a suitable prolongation that preserves constants in $V_{\mathrm {HOST}}$, compare Fig. 2. To overcome this issue in a way that is non-invasive also to AMG we simply set up an auxiliary matrix

$$\begin{aligned} \bar{K}_{\mathrm {H,H}} := \left( \begin{array}{cc} K_{\mathrm {HF,HF}} &{} \bar{K}_{\mathrm {HF,HF^\bot }} \\ \bar{K}_{\mathrm {HF^\bot ,HF}} &{} \bar{K}_{\mathrm {HF^\bot ,HF^\bot }} \\ \end{array} \right) \end{aligned}$$

where we exchange the matrix blocks that involve $\Phi _{\mathrm {HOST}}$ in $K_{\mathrm {H,H}}$ by their unweighted counterparts, i.e. which are computed using the classical FE basis functions $\phi _k$ instead of the products $\Phi _{\mathrm {HOST}} \phi _k$. The matrix block $K_{\mathrm {HF,HF}}$ is unchanged so that the non-invasive character of our approach to the HOST code is fully maintained and no additional assembly on the sub-domain $\Omega _{\mathrm {HOST, FT}}$ that is handled by the HOST code is necessary. Since $\bar{K}_{\mathrm {H,H}}$ now satisfies all assumptions of AMG, we obtain a suitable coarse space $V_{\mathrm {HOST},J-1}$ and associated prolongation $\bar{P}_{\mathrm {HOST}, J-1}$ from the application of AMG to $\bar{K}_{\mathrm {H,H}}$, compare Fig. 2. In fact, AMG automatically computes a sequence of coarse spaces $V_{\mathrm {HOST},i}$ and associated prolongations $\bar{P}_{\mathrm {HOST}, i-1}^i:V_{\mathrm {HOST},i-1} \rightarrow V_{\mathrm {HOST},i}$.

Fig. 2

Contour plots of the prolongation errors (left) attained for the constant function when using the weighted space $\Phi _{\mathrm {HOST}} V_{\mathrm {HOST}}$ in the AMG construction of the prolongation $P_{\mathrm {HOST}, J-1}^J$ for a small (top) and a large overlap region (bottom). Surface plots of the prolongation error of the smooth function $\sin (3\pi x)\cos (3\pi y)$ using $P_{\mathrm {HOST}, J-1}^J$ (center) and $\bar{P}_{\mathrm {HOST}, J-1}^J$ (right)

For the embedded space $V_{\mathrm {ENR}}$, which in our case is itself a PUM space, we construct a sequence of suitable prolongations $\bar{P}_{\mathrm {ENR}, i-1}^i:V_{\mathrm {ENR},i-1} \rightarrow V_{\mathrm {ENR},i}$ directly by so-called global-to-local $L^2$-projections [22, 23] based on a geometric coarsening process. Thus, we can now define a sequence of prolongations

$$\begin{aligned} P_{i-1}^i := \left( \begin{array}{cc} \bar{P}_{\mathrm {HOST}, i-1}^i &{} 0 \\ 0 &{} \bar{P}_{\mathrm {ENR}, i-1}^i \\ \end{array} \right) \end{aligned}$$

and respective coarser versions $K_{\mathrm {BND},i}$ of our overall stiffness matrix $K_{\mathrm {BND}}=K_{\mathrm {BND},J}$ by the so-called Galerkin operators

$$\begin{aligned} K_{\mathrm {BND},i-1} := (P_{i-1}^i)^T K_{\mathrm {BND}, i} P_{i-1}^i. \end{aligned}$$

Note, however, that this overall coarsening process in fact coarsens the two componentes $V_{\mathrm {HOST}}$ and $V_{\mathrm {ENR}}$ independently and thus may not be optimal on the overlap region $\Omega _\Phi =\Omega _{\mathrm {HOST}} \cap \Omega _{\mathrm {ENR}}$ where the two spaces interact substantially. Therefore, the overall performance of the proposed scheme will be affected somewhat by the absolute size of the overlap region $\Omega _\Phi$. Unlike in classical domain decomposition methods where larger overlap yields a convergence improvement, we, however, anticipate that an increasing overlap will rather worsen the convergence behavior since our independent coarsening process then ignores stronger interactions between the spaces.

As a final component we need to specify the approximate subspace solvers or smoothers on the resulting coarse spaces $V_{\mathrm {BND},i}$ to instantiate the iterations (15) and (17). In the following we focus on iterations of the form (17), in particular we employ the classical multigrid iteration $M^{\nu _1, \nu _2}_\gamma$ given in Algorithm 3.1 and consider different numbers of smoothing steps $\nu =\nu _1=\nu _2$ as well as the V-cycle ($\gamma =1$) and the W-cycle ($\gamma =2$).

As smoothers $S^{\mathrm {pre}}_l$ and $S^{\mathrm {post}}_l$ in Algorithm 3.1 we simply use Gauß–Seidel iterations, which provide acceptable error reduction but their smoothing effect seems to be somewhat less pronounced in $\Omega _{\Phi }$ for larger overlap regions $\Omega _{\Phi }$, see Fig. 3.

Fig. 3

Surface plots of iterates of a Gauß–Seidel smoother for iteration 1, 3, 5 (left to right) using a random initial guess on a blended discretization using a small (top) and a large overlap region (bottom)

Numerical results

In this section we present some results of our numerical experiments using the embedded enriched PUM within a classical FE simulation as discussed above. To this end, we consider a set of isotropic model problems focussing on the optimality of the proposed solver only. A detailed study of the solvers robustness with respect to varying material coefficients is the subject of future research. In particular, we are concerned with the approximation of the Poisson problem

$$\begin{aligned} \begin{array}{rcl} \displaystyle -\Delta u &{} = &{} f \quad \text { in } \Omega \subset \mathbb {R}^d,\\ \displaystyle u &{} = &{} g \quad \text { on } \Gamma _D \subset \partial \Omega ,\\ \displaystyle \frac{\partial u}{\partial n} &{} = &{} h\quad \text { on } \Gamma _N = \partial \Omega {\setminus } \Gamma _D,\\ \end{array} \end{aligned}$$

(19)

in two space dimensions on a square domain, see Fig. 1, where we embed a smooth enrichment space $V_{\mathrm {ENR}}$ to identify the best performance of our proposed solver. Then, we consider a non-convex domain, see Fig. 4, where we embed an enrichment space $V_{\mathrm {ENR}}$ that contains singular functions to efficiently resolve the corner singularity. Finally, we consider a linearly elastic bar in three space dimensions, see Fig. 5, i.e.

$$\begin{aligned} \begin{array}{rcll} -{\mathbf {div}} \varvec{\sigma }(\vec {u}) &{} = &{} 0 &{} \quad \text {in } \Omega \subset \mathbb {R}^d,\\ \vec {u} &{} = &{} \vec {g} &{} \quad \text {on } \Gamma _D \subset \partial \Omega ,\\ \varvec{\sigma }(\vec {u}) \cdot \vec {n} &{} = &{} 0 &{} \quad \text {on } \Gamma _C,\\ \varvec{\sigma }(\vec {u}) \cdot \vec {n} &{} = &{} \vec {h} &{} \quad \text {on } \Gamma _N = \partial \Omega {\setminus } \Gamma _D,\\ \end{array} \end{aligned}$$

(20)

with the stress tensor $\varvec{\sigma }(\vec {u}) := \mathcal {C} \varvec{\varepsilon }(\vec {u}) = 2\mu \varvec{\varepsilon }(\vec {u}) + \lambda {\text {trace}}(\varvec{\varepsilon }(\vec {u})) \mathbb {I}$ and the infinitesimal strain tensor $\varvec{\varepsilon }(u):=\frac{1}{2}(\varvec{\nabla } \vec {u} + (\varvec{\nabla } \vec {u})^T)$, to study the performance of our proposed solver for systems of equations in higher dimensions. Moreover, the selected model problems employ different embedding configurations with respect to the intersection of $\Omega _{\mathrm {ENR}}$ with the boundaries $\Gamma _D$ and $\Gamma _N$ of the global simulation domain $\Omega$, compare Figs. 1, 4 and 5.

Fig. 4

Sketch of a non-convex domain $\Omega$ in two dimensions depicting the employed partitioning into $\Omega _{\mathrm {HOST, FT}}$ (drak gray), $\Omega _\Phi =\Omega _{\mathrm {HOST}}\cap \Omega _{\mathrm {ENR}}$ (light gray), and $\Omega _{\mathrm {ENR, FT}}$ (white). The boundary data for this model problem is given by $g=0$ on $\Gamma _D$ and $h=\nabla \Big ((x^2 + xy + y^2 + 1)(r^\frac{2}{3} \sin (\frac{2\theta - \pi }{3}))\Big ) \cdot n$ on $\Gamma _N$ where n denotes the outer normal

Fig. 5

Sketch of a pre-cracked bar in three dimensions depicting the partitioning of the domain $\Omega$ into $\Omega _{\mathrm {HOST, FT}}$ (drak gray), $\Omega _\Phi =\Omega _{\mathrm {HOST}}\cap \Omega _{\mathrm {ENR}}$ (light gray), and $\Omega _{\mathrm {ENR, FT}}$ (white). The boundary data for this model problem is given by $\Gamma _D = \Gamma _L \cup \Gamma _R$, $g=(0, 0, 0)^T$ on $\Gamma _L$, $g=(0.02, 0, 0)^T$ on $\Gamma _R$ and $h=(0, 0, 0)^T$ on $\Gamma _N=\partial \Omega \setminus \Gamma _D$

In all our experiments we consider discretizations which satisfy our assumption (8) by choosing the support sizes of basis functions in $V_{\mathrm {HOST}}$ and $V_{\mathrm {ENR}}$ of comparable size, see Fig. 6, and use (bi/tri-)linear finite elements in $V_{\mathrm {HOST}}$ and linear polynomials with additional enrichments in $V_{\mathrm {ENR}}$. We employ the proposed multilevel iteration as a stand-alone solver as well as a preconditioner for the conjugate gradient method (CG) and measure the number of iterations required to reduce the initial residual by ten orders of magnitude. From an optimality point of view, we are mostly interested in obtaining iteration numbers n that are independent of the number of employed levels k. From the measured iteration numbers given in Table 1, we see that the V-cycle stand-alone solver with only a single pre- and post-smoothing step already provides acceptable iteration numbers $n_\mathrm {V(1,1)}< 45$ which, however, are not completely independent of the number of employed levels k. Yet, increasing the number of smoothing steps to $\nu =3$ or changing to the more expensive W-cycle yields constant iteration numbers $n_\mathrm {V(3,3)}$ and $n_\mathrm {W(1,1)}$ independent of k, see also Fig. 7. In fact, further experiments with the proposed multilevel iteration showed that it is already sufficient to increase the number of smoothing steps on coarser levels only. Thus, indicating that the quality of the corrections from coarser levels obtained in a V(1, 1)-cycle is somewhat deminished for larger k. Nevertheless, the use of the V(1, 1)-cycle as a preconditioner in CG yields fairly stable iteration numbers $n_\mathrm {CGV(1,1)} < 20$ and provides the fastest time-to-solution for the considered numbers of levels k. Note that the results summarized in Table 1 were obtained with a small overlap region $\Omega _\Phi$ of a single element on the finest level; i.e. the overlap region is in fact shrinking as we refine the discretization. As mentioned above, we anticipate that for a larger overlap region $\Omega _\Phi$ with fixed volume for all levels k, i.e.  an increasing number of elements in the overlap as we refine, the convergence behavior of the proposed solver will deteriorate somewhat. In fact, the results given in Table 2 show that the number of iterations increases not only but also grows with larger number of levels k even when we use the rather expensive W(1, 1)-cycle as a preconditioner in CG. Thus, the results confirm our expectation that it is advisable to choose an overlap region $\Omega _\Phi$ whose diameter is proportational to the meshwidth on the finest level employed (unlike in classical domain decomposition approaches) since such a choice yields the least amount of work in the assembly of the blended linear system $K_{\mathrm {BND}}$ and it gives the best solver performance.

Fig. 6

Sketch of the blended function discretization space depicting the supports of a basis function $\phi _{\mathrm {HOST}} \in V_{\mathrm {HOST}}$ (light gray) and $\phi _{\mathrm {ENR}} \in V_{\mathrm {ENR}}$ (dark gray). The support sizes are of comparable size throughout the paper; i.e. ${\text {diam}}({\text {supp}}(\phi _{\mathrm {HOST}})\sim {\text {diam}}({\text {supp}}(\phi _{\mathrm {ENR}})$

Table 1 Measured iteration numbers n for the stand-alone solver and $n_{CG}$ for the preconditioned CG method attained for the model problem (19) on the configuration depicted in Fig. 1 using a single element overlap

Fig. 7

Convergence histories for the V(1, 1)-cyle as a stand-alone solver (left) and the W(1, 1)-cycle as a preconditioner in CG (right)

Table 2 Measured iteration numbers $n_{CG}$ for the preconditioned CG method attained for the model problem (19) on the configuration depicted in Fig. 1 using a fixed volume overlap

Now that we have in principle identified the best attainable convergence behavior of the proposed method for a smooth problem employing a smooth enrichment space $V_{\mathrm {ENR}}$, let us consider a more relevant case where $V_{\mathrm {ENR}}$ contains problem-dependent singular functions. To this end, we consider (19) on a non-convex domain, see Fig. 4. Here, the discretization space $V_{\mathrm {ENR}}$ on the region $\Omega _{\mathrm {ENR}}$ is defined as

$$\begin{aligned} V_{\mathrm {ENR}} = \sum _{i=1}^N \varphi _i \Big ({\mathcal {P}}_1 + {\text {span}}\langle r^{\frac{2}{3}} \sin \big (\frac{2\theta -\pi }{3}\big )\rangle \Big ) \end{aligned}$$

(21)

where $(r, \theta )$ denote polar coordinates with respect to the re-entrant corner of $\Omega$. Thus, we employ enrichment by a singular function everywhere in $\Omega _{\mathrm {ENR}}$, see [24] for details on the construction of a stable basis for $V_{\mathrm {ENR}}$. Moreover, we use so-called block-Gauß–Seidel relaxation in $V_{\mathrm {ENR}}$ where we collect all degrees of freedom defined on the same patch into a single block, see [22, 23] for details. The attained number of iterations are given in Table 3. The overall number of iterations is somewhat larger than in the previous case, however, the number of iterations are essentially independent of the number of levels. It is also noteworthy to point out, that in this model configuration the V(3, 3)-cycle substantially outperforms the W(1, 1)-cycle which shows the improved smoothing property of the patch-based block-Gauß–Seidel relaxation in $V_{\mathrm {ENR}}$. Nevertheless, the fastest time-to-solution for the considered discretizations with more than 13 million degrees of freedom is still obtained by CG preconditioned by V(1, 1)-cycle which of course also benefits from the improved smoothing property.

Table 3 Measured iteration numbers $n_{CG}$ for the preconditioned CG method attained for the model problem (19) on the configuration depicted in Fig. 4 using a single element overlap

Finally, we consider the system of partial differential equations (20) in three space dimensions. Here, we employ a so-called point-based AMG approach [21] for the coarsening in $\Omega _{\mathrm {HOST}}$ and utilize one more layer of bock-partitioning which collects all three displacement components in the Gauß–Seidel smoother, i.e. we now use a $(3\times 3)$-block relaxation in $V_{\mathrm {HOST}}$ combined with the block-relaxation in $V_{\mathrm {ENR}}$ described above. We discretize (20) with trilinear elements in $V_{\mathrm {HOST}}$ and use singular and discontinuous enrichments for the treatment of the crack in $V_{\mathrm {ENR}}$ (besides linear polynomials). The performance of our proposed solver is summarized in Table 4. Again, we find essentially constant iteration numbers for CG preconditioned by W(1, 1)-cycle and slightly increasing iteration numbers when using a V-cycle preconditioner. Yet, the fastest time-to-solution is still attained when using the V(1, 1)-cycle as preconditioner.

Table 4 Measured iteration numbers $n_{CG}$ for the preconditioned CG method attained for the model problem (20) on the configuration depicted in Fig. 5 using a single element overlap

In summary, the presented results clearly show that the proposed solver yields close to optimal convergence in two and three dimensions when using a small overlap. Using a small overlap is moreover beneficial to the total computational cost in the assembly of the blended linear system and still yields optimal approximation errors.

Concluding remarks

In this paper we proposed a constructive non-invasive approach to the design of efficient multilevel solvers for embedded enriched approximations. The non-invasive embedding scheme is based on a partition of unity approach and can essentially blend arbitrary (overlapping) approximation spaces, yet we consider the special case of embedding an enriched partition of unity space into a classical finite element space. The proposed solver utilizes non-invasive algebraic multigrid technology [16,17,18,19,20] for the automatic construction of a sequence of coarser sub-spaces of the employed finite element space and a sequence of enriched partiton of unity spaces obtained via a geometric coarsening scheme [22]. The presented results clearly indicate that the proposed method can attain (close to) optimal convergence behavior when a small overlap or blending region is employed. A detailed study on the optimal selection of parameters and robustness properties of the proposed scheme is the subject of ongoing and future research.