# CBS Constants & Their Role in Error Estimation for Stochastic Galerkin Finite Element Methods

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## Abstract

Stochastic Galerkin finite element methods (SGFEMs) are commonly used to approximate solutions to PDEs with random inputs. However, the study of a posteriori error estimation strategies to drive adaptive enrichment of the associated tensor product spaces is still under development. In this work, we revisit an a posteriori error estimator introduced in Bespalov and Silvester (SIAM J Sci Comput 38(4):A2118–A2140, 2016) for SGFEM approximations of the parametric reformulation of the stochastic diffusion problem. A key issue is that the bound relating the true error to the estimated error involves a CBS (Cauchy–Buniakowskii–Schwarz) constant. If the approximation spaces associated with the parameter domain are orthogonal in a weighted \(L^2\) sense, then this CBS constant only depends on a pair of finite element spaces \(H_{1}, H_{2}\) associated with the spatial domain and their compatibility with respect to an inner product associated with a parameter-free problem. For fixed choices of \(H_{1}\), we investigate non-standard choices of \(H_{2}\) and the associated CBS constants, with the aim of designing efficient error estimators with effectivity indices close to one. When \(H_1\) and \(H_2\) satisfy certain conditions, we also prove new theoretical estimates for the CBS constant using linear algebra arguments.

## Keywords

Error estimation for stochastic PDE problems Stochastic finite element method Stochastic Galerkin method Strengthened Cauchy–Schwarz inequality CBS constants## Mathematics Subject Classification

35R60 60H35 65N30 65F10## 1 Introduction

The motivation for this work is the design of efficient a posteriori error estimators for adaptive Galerkin finite element approximation of solutions to partial differential equations (PDEs). In particular, we are interested in PDEs with random inputs and so-called stochastic Galerkin finite element methods (SGFEMs) (see [4, 5, 14, 22, 24, 31]). When the inputs are represented by a finite, or countably infinite number of random variables \(\xi _{m}: \varOmega \rightarrow \mathbb {R}\), \(m=1,2,\ldots ,\) where \(\varOmega \) is a sample space, it is conventional to reformulate the stochastic problem of interest as a high-dimensional deterministic one, whose solution depends on a set of *parameters* \(y_{m}=\xi _{m}(\omega )\). Unlike sampling methods such as Monte Carlo, in the SGFEM approach, an approximation is sought in a tensor product space of the form \(H_1 \otimes P\) where \(H_1\) is an appropriate finite element space associated with the spatial domain and *P* is a set of polynomials associated with the parameter domain. When the number of active parameters is high, the dimension of \(H_1 \otimes P\) for standard choices of \(H_1\) and *P* can become unwieldy. To remedy this, we can either work with standard approximation spaces and deal with the resulting very large discrete systems by using smart linear algebra techniques (see [20, 21, 25, 26, 32, 33]), or we can use an adaptive approach, starting in a low-dimensional space \(H_1^{0} \otimes P^{0}\), and using a posteriori error estimators to decide whether it is necessary to enrich \(H_1^{0}\) or \(P^{0}\), or both. This allows us to build up a tailored sequence of approximation spaces \(H_1^{\ell } \otimes P^{\ell }, \ell =0,1, \ldots \) incrementally, so that the dimension of the final space is balanced against an error tolerance for a quantity of interest.

### Assumption 1

The parameter-free function \(a_0(\mathbf {x})\) typically represents the mean with each term \(a_m(\mathbf {x})y_m\) representing a perturbation away from the mean, while (5) helps ensure the well posedness of the weak formulation of (1a)–(1b).

Next, we recall the classical strengthened Cauchy–Buniakowskii–Schwarz (CBS) inequality (see [1, Theorem 5.4]), a key tool in many areas of numerical analysis.

### Theorem 1

*H*be a Hilbert space equipped with inner product \((\cdot ,\cdot )\) and induced norm \(||\cdot ||\) and let

*U*,

*V*be a pair of finite-dimensional subspaces of

*H*satisfying \(U\cap V=\{0\}\). Then, there exists a constant \(\gamma \in [0,1)\), depending only on

*U*and

*V*such that

The design of error estimators for SGFEMs for parameter-dependent PDEs is still under development. However, there have been a few recent works (see [8, 9, 10, 11, 12, 16, 17, 18, 29]) for the model problem (1a)–(1b). In [12] and [11], algorithms constructing so-called *sparse* SGFEM approximations are driven by a priori error analysis, where the error associated with each discretisation parameter is balanced against the total number of degrees of freedom. In [16] and [17] a general framework for an explicit residual-based error estimation strategy for SGFEMs is proposed, where the selection of hierarchical approximation spaces is driven by a Dörfler marking strategy [15] on both the spatial and parameter domains. In both works, overestimation up to a factor of 10 of the true error is reported. In [18] a similar approach is taken, where residuals are computed using an equilibrated fluxes strategy and overestimation up to a factor of 5 is reported.

We focus on the (implicit) approach taken in [8, 9, 10] which is based on solving local subproblems for the error over a ‘detail’ space. The bound for the effectivity index of the resulting error estimators depends on a CBS constant. In [8, 9, 10] no insight into which choices of detail space result in a sharp error bound (effectivity indices close to one) is given. In this paper we provide that analysis. Specifically, we provide detailed information about the CBS constant and derive theoretical bounds for it, for certain choices of SGFEM solution and detail spaces. Due to the way that the spaces associated with the parameter domain are chosen, the CBS constants needed to analyse the estimators in [8, 9, 10] depend only on a pair of finite element spaces on the spatial domain. Hence, our results are also applicable to the design of adaptive finite element schemes for deterministic PDEs. We investigate which choices of detail FEM space result in CBS constants close to zero, to help ensure a sharp error bound. In particular, due to cost restrictions imposed to avoid high-dimensional detail spaces, we investigate non-standard choices that aren’t typically considered in the deterministic setting. The error estimation strategy in [29] also relies on a CBS constant, but in a different setting. Enrichment of the finite element space is not considered.

### 1.1 Outline

In Sect. 2, for the benefit of readers who are not familiar with the area, we review classical results from [1, 6] concerning a posteriori error estimation for Galerkin approximation. In Sect. 3 we demonstrate how those results are applied to SGFEM approximations of (1a)–(1b), leading to a simplified analysis of the error estimator introduced in [10] and the associated error bound. In particular, we show how the bound depends on a CBS constant associated with two finite element spaces \(H_{1}\) and \(H_{2}\). In Sect. 4 we first remind the reader how to compute CBS constants numerically. We then study some (non-standard) pairs of \(H_{1}\) and \(H_{2}\) and compute the associated CBS constants. In Sect. 5 we demonstrate that if \(H_1\) is the space of piecewise bilinear (\(\mathbb {Q}_{1}\)) functions, theoretical estimates for the CBS constant can be obtained using a novel linear algebra approach for several choices of \(H_2\). Finally, in Sect. 6 we present numerical results demonstrating the quality of the aforementioned error estimator, and the vital importance of choosing the right detail spaces.

## 2 Classical a Posteriori Error Estimation

*V*be a Hilbert space with norm \(||\cdot ||_V\) and let \(B:V\times V\rightarrow \mathbb {R}\) and \(F:V\rightarrow \mathbb {R}\) denote a bounded and coercive bilinear form and linear functional, respectively. Consider the problem:

*V*with the ‘energy’ norm

*X*be an \(N_X\)–dimensional subspace of

*V*. We then solve:

*e*satisfies the problem:

*W*is richer than

*X*) and consider the following problem:

*W*. We deduce then that the function \(e_W\in W\) satisfying (12) estimates the true error \(e\in V\) satisfying (10). Whilst we do not compute \(u_W\), it is clear that the quality of that Galerkin approximation (and hence the choice of

*W*) determines the quality of the estimator \(e_W\). To analyse this, we require the following assumption.

### Assumption 2

In many applications, Assumption 2 is reasonable (see [1, p. 88]). The relationship between \(||e||_B\) and \(||e_W||_B\) is summarised in the next result.

### Theorem 2

The interpretation of (14) is as follows; \(||e_W||_B\) will never overestimate the true error \(||e||_B\), but could underestimate it by a factor of \((1-\beta ^2)^{-1/2}\).

*W*is more convenient to work with. We may then consider the alternative problem:

### Theorem 3

### Theorem 4

If *X* and *Y* are orthogonal with respect to the inner product \(B_0(\cdot ,\cdot )\) then \(\gamma = 0\) and \(||e_Y||_{B_0} = ||e_0||_{B_0}\). Consolidating Theorems 2–4 yields the final result.

### Theorem 5

In summary, the quality of the energy error estimate \(||e_Y||_{B_0}\) is determined by two constants \(\beta \) and \(\gamma \), which both depend on *X* and *Y*. Ideally, we want \(\sqrt{1-\beta ^2}\sqrt{1-\gamma ^2} \approx 1\). Given a fixed initial approximation space *X*, what is the best choice of detail space *Y*, from the point of view of obtaining the best possible error estimate? This is the essence of our investigation.

## 3 The Parametric Diffusion Problem

*V*is equipped with the norm \(|| \cdot ||_V\), where \(||v||_V^2 = \int _{\varGamma }||v(\cdot ,\mathbf {y})||_{H_0^1(D)}^2\ d\pi (\mathbf {y})\) and the bilinear form \(B:V\times V\rightarrow \mathbb {R}\) and the linear functional \(F:V\rightarrow \mathbb {R}\) are given by

### Assumption 3

### 3.1 SGFEM Approximation

We now seek a Galerkin approximation to \(u\in V\) satisfying (23). As in Sect. 2, we denote by *X* an \(N_X\)–dimensional subspace of *V*. Here, we exploit the tensor product structure of *V* and choose \(X:=H_1\otimes P\), where \(H_1\subset H_0^1(D)\) and \(P\subset L^2_{\pi }(\varGamma )\). We then look for \(u_X\in X\) satisfying (9).

*D*and \(P=\text {span}\{\varphi _{i}(\mathbf {y})\}_{i=1}^s\) to be a space of global (multivariate) polynomials on \(\varGamma \), so that \(N_{X}=ns\). We choose the basis functions for

*P*to be orthonormal with respect to \(\langle \cdot ,\cdot \rangle _{L_{\pi }^2(\varGamma )}\). To this end, we introduce the set of finitely supported multi-indices; \(J:=\{\mu = (\mu _1,\mu _2,\dots )\in \mathbb {N}_0^{\mathbb {N}};\ \#\ \text {supp}(\mu ) < \infty \}\), where \(\text {supp}(\mu ) :=\{m\in \mathbb {N};\ \mu _m\ne 0\}\). For a given multi-index \(\mu \in J\) we then construct

*P*is equivalent to choosing a set of multi-indices \(J_P\subset J\) with cardinality \(\text {card}(J_P)=s\).

To compute a Galerkin approximation \(u_{X} \in X\) satisfying (9), it is essential that the sum in (26) has a finite number of nonzero terms. It is not necessary to truncate the diffusion coefficient a priori. We need only assume that *P* contains polynomials in which a *finite* number of parameters \(y_{m}\) are ‘active’. If we assume that the first *M* parameters are active, then, provided (2) holds, \(B_m(u_{X},v) = 0\) for \(u_{X}, v \in X\) for all \(m >M\) (e.g., see [8]). In other words, the projection onto \(X=H_{1} \otimes P\) truncates the sum after *M* terms.

### 3.2 A Posteriori Error Estimation

Suppose we now choose a second SGFEM space \(W\subset V=H_{0}^{1}(D) \otimes L_{\pi }^{2}(\varGamma )\) such that \(W \supset X:=H_{1} \otimes P\) and solve (12) to obtain an estimator \(e_{W} \in W\) for the error \(e=u-u_{X}\). If Assumption 2 holds for the chosen spaces *X* and *W*, then (14) also holds, where \(||\cdot ||_B \) is the energy norm induced by the bilinear form defined in (24). In addition, due to the norm equivalence (28), the bound (17) also holds, where \(e_{0} \in W\) satisfies (15) and \(||\cdot ||_{B_0}\) is the norm induced by the bilinear form defined in (27a).

*W*. Following [10], we choose

*Q*. If \(J_P\cap J_Q = \emptyset \), then we have \(P\cap Q = \{0\}\) as required. In this case,

*P*and

*Q*are mutually orthogonal with respect to \(\langle \cdot ,\cdot \rangle _{L_{\pi }^2(\varGamma )}\) since

*Y*, we can then compute the error estimate \(\eta := ||e_Y||_{B_0}\) by solving (19) and the bound (21) holds. Combining all these results yields the result of Theorem 5. For completeness, we restate this for our parametric diffusion problem.

### Theorem 6

*Q*and

*Y*as in (31). Let \(e_Y \in Y\) satisfy (19). If Assumptions 1–3 hold, then \(\eta := ||e_Y||_{B_0}\) satisfies

*Y*is chosen as in (31), problem (19) decouples. Since \(Y_1 \cap Y_2 = \{0\}\), \(e_Y = e_{Y_1} + e_{Y_2}\) for some \(e_{Y_1}\in Y_1\), \(e_{Y_2}\in Y_2\) and thus \(B_0(e_Y,v) = B_0(e_{Y_1}+e_{Y_2},v) = B_0(e_{Y_1},v) + B_0(e_{Y_2},v)\) for all \(v\in Y\). By choosing test functions \(v \in Y_{1}\) and \(v \in Y_{2}\) in (19) and considering the identity (33), we find that \(e_Y \in Y\) satisfying (19) can be determined by solving the lower-dimensional problems

*P*and

*Q*, and the decoupling of (15) into two smaller problems over \(((H_1 \oplus H_2)\otimes P)\) and \((H_2\otimes Q)\). A CBS constant is introduced into the analysis by splitting the former into \(H_1\otimes P\) and \(H_2\otimes P\). Our approach is subtly different. We introduce a CBS constant by splitting the augmented space

*W*into

*X*and

*Y*, as would be done for the analogous deterministic problem (for which \(X = H_1\) and \(Y = H_2\)).

*P*and

*Q*are orthogonal with respect to \(\left<\cdot , \cdot \right>_{L_{\pi }^{2}}\), \(Y_{1}\) and \(Y_{2}\) are orthogonal with respect to \(B_{0}(\cdot , \cdot )\) and so \(B_0(u,v) = B_0(u,v_1)\) for all \(u \in X\) and \(v \in Y\). Hence, using \(||v_1||_{B_0}^2 = ||v||^2_{B_0} - ||v_2||^2_{B_0}\), we have \(|B_0(u,v)| \le \gamma ||u||_{B_0}||v_1||_{B_0} \le \gamma ||u||_{B_0}||v||_{B_0}\), for all \(u\in X,\) and \(v\in Y\), where \(\gamma \in [0,1)\) is the

*same*constant satisfying (39). Consequently, \(\gamma \) in (34) can be determined by analyzing the spaces \(H_1\) and \(H_2\).

*P*and

*Q*do not play a role. They do, of course, affect the saturation constant \(\beta \), and this will be discussed in Sect. 6.

### 3.3 Estimated Error Reductions

### Theorem 7

Given \(H_{2}\) and *Q*, Theorem 7 allows us to assess whether enrichment of \(H_1\) (with functions in \(H_2\)) is more beneficial than enrichment of *P* (with functions in *Q*). We may choose \(X^* = W_1\) or \(X^* = W_2\). Our choice is determined by which space offers the greatest estimated error reduction per additional degree of freedom. Note that the bound (44) is independent of the saturation constant \(\beta \), and choosing \(H_2\) in (31) so that the constant \(\gamma \) in (39) is small tightens the bound (44). That is, if the CBS constant is small, we can have more confidence in our decisions when performing adaptivity. We now study this constant for various choices of \(H_1\) and \(H_2\).

## 4 Numerical Estimates of CBS Constants

The constant \(\gamma \in [0, 1)\) in (34) and (44), which is equivalent to the constant \(\gamma \in [0, 1)\) satisfying (39), is not unique. Given \(H_1\) and \(H_2\), we want to find the smallest such constant, the CBS constant. We now recall a standard result from [19] which leads to a numerical method for computing the CBS constant associated with (39).

*M*has a particular block structure, Theorem 1 along with

*U*and

*V*in (46), leads to the following result (see [19] for a proof).

### Corollary 1

The matrix \(M\in \mathbb {R}^{N\times N}\) is symmetric and positive definite and has the structure (47) with \([A]_{ij} = \langle \phi _{i}, \phi _{j}\rangle \), for \(i,j=1,\ldots ,n\), \([B]_{ij} = \langle \psi _{i}, \psi _{j}\rangle \) for \(i,j=1,\ldots , m\) and \([C]_{ij} = \langle \phi _{i}, \psi _{j}\rangle \) for \(i=1,\ldots , n\) and \(j=1,\ldots , m\). By Corollary 1, there exists a constant \(\gamma \in [0,1)\) such that (48) holds, which is equivalent to (50). Therefore, the CBS constant \(\gamma _{\min }\) satisfying (50) can be computed numerically by solving the eigenvalue problem (49).

*D*, we construct \(H_2\) element-wise. That is, we insist that \(H_2\) admits the decomposition

*h*, and estimate the CBS constant by solving the eigenvalue problem (49).

### Example 1

*h*. Now let \(H_1\) be the space of continuous functions that are piecewise bilinear on \(\mathcal {T}_h\) (denoted \(H_{1}=\mathbb {Q}_1(h)\)). On each \(\square _{k}\) we construct a local space \(H_{k,2}\) of dimension \(m_{k} \le 5\), by defining bubble functions at the edge midpoints and the element centroid (the \(\mathbb {Q}_{1}\) nodes that would be introduced by a uniform mesh refinement). We consider the following options. The name given to the resulting space \(H_{2}\) is shown in brackets.

- 1.
*Biquadratic bubble functions*(\(\mathbb {Q}_{2}(h)\)) Consider the standard set of nine biquadratic (\(\mathbb {Q}_{2}\)) element basis functions and keep only those associated with the five selected nodes. - 2.
*Biquartic bubble functions*(\(\mathbb {Q}_{4}(h)\)) Consider the standard set of twenty-five biquartic (\(\mathbb {Q}_{4}\)) element basis functions and keep only the desired five. - 3.
*Piecewise bilinear bubble functions*(\(\mathbb {Q}_{1}(h/2)\)) Subdivide each element into four smaller ones of size*h*/ 2, and concatenate the standard \(\mathbb {Q}_{1}\) basis functions associated with each new element at the five chosen nodes. - 4.
*Piecewise biquadratic bubble functions*(\(\mathbb {Q}_{2}(h/2)\)) This is the same as option 3 but with \(\mathbb {Q}_{2}\) basis functions.

*h*. In [8], the authors choose \(H_{2}=\mathbb {Q}_{1}(h/2)\) to define the error estimator \(\eta =\Vert e_{Y} \Vert _{B_{0}}\) described in Sect. 3 when \(u_{X}\) is computed with \(H_{1}=\mathbb {Q}_{1}(h)\). Hence, when \(a_{0}=1\), the associated CBS constant is \(\gamma _{\min }\le \sqrt{0.375}= \sqrt{3/8}\). However, of the four choices considered, \(H_{2}=\mathbb {Q}_{4}(h)\) yields the smallest CBS constant.

Computed values of \(\gamma _{\min }^2\) for Example 1, for varying *h*. \(H_1\) is the usual \(\mathbb {Q}_{1}\) finite element space and four choices of \(H_{2}\) are considered

Mesh | | | \(\mathbb {Q}_{2}(h)\) | \(\mathbb {Q}_{4}(h)\) | \(\mathbb {Q}_{1}(h/2) \) | \(\mathbb {Q}_{2}(h/2) \) |
---|---|---|---|---|---|---|

\(4\times 4\) | \(2^{-1}\) | 73 | 0.4106 | 0.0109 | 0.3381 | 0.0401 |

\(8\times 8\) | \(2^{-2}\) | 337 | 0.4454 | 0.0119 | 0.3673 | 0.0437 |

\(16\times 16\) | \(2^{-3}\) | 1441 | 0.4527 | 0.0121 | 0.3735 | 0.0445 |

\(32\times 32\) | \(2^{-4}\) | 5953 | 0.4541 | 0.0121 | 0.3747 | 0.0446 |

\(64\times 64\) | \(2^{-5}\) | 24193 | 0.4544 | 0.0121 | 0.3749 | 0.0446 |

Converged value | 0.4545 | 0.0121 | 0.3750 | 0.0446 |

### Example 2

- 1.
*Biquartic bubble functions*(\(\mathbb {Q}_{4}(h)\)) Consider the set of twenty-five \(\mathbb {Q}_{4}\) element basis functions associated with \(\square _{k}\) but retain only those associated with the nodes indicated by the clear and grey markers in Fig. 2. - 2.
*Piecewise biquadratic bubble functions*(\(\mathbb {Q}_{2}(h/2)\)) Subdivide each element into four smaller ones of size*h*/ 2, and concatenate the standard \(\mathbb {Q}_{2}\) basis functions associated with the new elements at the nodes indicated by the clear and grey markers in Fig. 2.

In Table 2 we record \(\gamma _{\min }^2\) for each choice of \(H_2\) for varying *h*. In [10], the authors choose \(H_{2}=\mathbb {Q}_{4}^{r}(h)\) to define the error estimator \(\eta =\Vert e_{Y} \Vert _{B_{0}}\) described in Sect. 3, when \(u_{X}\) is computed with \(H_{1}=\mathbb {Q}_{2}(h)\). When \(a_{0}=1\), the CBS constant is \(\gamma _{\min }\le \sqrt{0.36}\). Of the four spaces considered, \(H_{2}=\mathbb {Q}_{4}^{r}(h)\) yields the smallest CBS constant.

Computed values of \(\gamma _{\min }^2\) for Example 2, for varying *h*. \(H_1\) is the usual \(\mathbb {Q}_{2}\) finite element space and four choices of \(H_{2}\) are considered

Mesh | | | \(\mathbb {Q}_{4}(h)\) | \(\mathbb {Q}_{2}(h/2)\) | \(N_r\) | \(\mathbb {Q}_{4}^{r}(h)\) | \(\mathbb {Q}_{2}^{r}(h/2)\) |
---|---|---|---|---|---|---|---|

\(2\times 2\) | \(2^{-0}\) | 57 | 0.3834 | 0.6764 | 41 | 0.3208 | 0.4904 |

\(4\times 4\) | \(2^{-1}\) | 273 | 0.4341 | 0.6911 | 209 | 0.3565 | 0.5579 |

\(8\times 8\) | \(2^{-2}\) | 1185 | 0.4391 | 0.6911 | 929 | 0.3595 | 0.5723 |

\(16\times 16\) | \(2^{-3}\) | 4929 | 0.4399 | 0.6911 | 3905 | 0.3599 | 0.5758 |

\(32\times 32\) | \(2^{-4}\) | 20097 | 0.4401 | 0.6911 | 16001 | 0.3600 | 0.5766 |

Converged value | 0.4401 | 0.6911 | 0.3600 | 0.5769 |

### 4.1 Local CBS Constants

*N*is large is not practical. Alternatively, we may derive a small eigenvalue problem associated with a single element. For all \(u\in H_2\) and \(v\in H_1\) we have

### Corollary 2

*U*and

*V*in (46), but for \(n_k\) and \(m_k\) in place of

*n*and

*m*. Then, there exists a constant \(\gamma _k\in [0,1)\) such that

*h*or \(a_{0}|_{k}\). To estimate \(\gamma _{\min }\), we only need to compute \(\gamma _{k,\min }\) for a single internal element (as this is larger than the constant associated with corner/edge elements).

### Remark 1

## 5 Theoretical Estimates of the CBS Constant

In this section we fix \(H_{k,1}\) to be the local \(\mathbb {Q}_{1}\) finite element space so that \(A_k\) is given by (58) and assume that the degrees of freedom are numbered as shown in Fig. 3. We also assume that \(H_{k,2}\) is chosen so that dim\((H_{k,2})=5\) and the resulting matrices \(B_k\) and \(C_k\) have a particular structure. Exploiting this structure, and using only linear algebra arguments, we show that the local CBS constant \(\gamma _{k,\min }\) can be calculated analytically without assembling and solving (57). To simplify notation, we drop the subscript *k*.

### Theorem 8

*A*is the \(\mathbb {Q}_1\) element stiffness matrix defined in (58). If the matrix \(CB^{-1}C^{\top }\in \mathbb {R}^{4\times 4}\) is of the form

### Proof

*A*are circulant matrices with zero row sums, we have

*B*is invertible, (62) gives \(0 = \mathbf {v}_2^TC\mathbf {u}_1 = -(C^T\mathbf {v}_2)^TB^{-1}(C^T\mathbf {v}_2)\) and \((B\mathbf {u}_1)^TB^{-1}(B\mathbf {u}_1)=0.\) Since \(B^{-1}\) is also invertible, we conclude that \(B\mathbf {u}_1=\mathbf {0}\) and \(\mathbf {u}_1 = \mathbf {0}\). Finally, \(\mathbf {u}_1 = \mathbf {0}\) and (62) gives \(C^T\mathbf {v}_2 = \mathbf {0}\). \(\square \)

### Theorem 9

### Proof

*A*with eigenvalue \(\theta =1\). Thus

\(\square \)

The next results show that if the matrices *B* and *C* have certain structures, then \(CB^{-1}C^{\top }\) always has the structure (60) and an explicit expression is available for the constant \(\beta \) in (65), and hence the CBS constant.

### Lemma 1

### Proof

*B*has the form (69) then so does \(B^{-1}\). We have

*C*gives

Combining the last two results, we see that to compute the CBS constant, we only need to know \(\alpha \) (one entry of *C*) and \(\alpha _{1}\) and \(\alpha _{3}\) (two entries of the first column of \(B^{-1}\)). We can determine the latter analytically by exploiting the spectral decomposition for circulant matrices. Lemma 2 is standard (for example, see [13]) and we apply it to \(\bar{B}\) in (69) in Lemma 3.

### Lemma 2

*D*are

### Lemma 3

*B*be given by

### Proof

Since \(\bar{B}\) is circulant, its eigenpairs are given by (72). Here, \(n=4\) and we have \(\omega _1 = \text {i}, \omega _2 = -1, \omega _3=-\text {i}\), and \(\omega _4=1\). The decomposition is standard (see [34, Corollary 5.16]). \(\square \)

Combining the above results, gives the following final result.

### Theorem 10

### Proof

*B*is symmetric and positive definite, so is \(\bar{B}\). Consequently, \(\lambda _{1}>0\) and \(\beta =4 \alpha ^{2} \left( b_1-b_3\right) ^{-1} >0\). The result follows by Theorem 9. \(\square \)

*B*and

*C*associated with the examples in Sect. 4.1 (corresponding to the four choices of \(H_{k,2}\) from Example 1)

*all*have the desired structure. The associated values of \(\alpha , b_{1}\) and \(b_{3},\) and the squares of the CBS constants are recorded in Table 3 (to stress that these are local quantities we reintroduce the subscript

*k*). The results match the numerical estimates obtained in Sect. 4.1. With the new approach, the matrices

*A*,

*B*and

*C*do not need to be assembled, and no eigenvalue problem needs to be solved.

The constants \(\alpha ,b_1,b_3 \in \mathbb {R}\) required to compute \(\gamma _{k,\min }^2=8 \alpha ^{2}(b_{1}-b_{3})\), when \(H_{k,1}\) is the local \(\mathbb {Q}_{1}\) space and \(H_{k,2}\) is chosen as in Example 1

\(H_{k,2}\) | \(\alpha \) | \(b_1\) | \(b_3\) | \(\gamma ^2_{k,\min }\) |
---|---|---|---|---|

\(\mathbb {Q}_{2}(h)\) | 1 / 3 | 88 / 45 | 0 | 0.4545 |

\(\mathbb {Q}_{4}(h)\) | 1 / 15 | 373 / 127 | 1 / 2326 | 0.0121 |

\(\mathbb {Q}_{1}(h/2)\) | 1 / 4 | 4 / 3 | 0 | 0.3750 |

\(\mathbb {Q}_{2}(h/2)\) | 1 / 12 | 56 / 45 | 0 | 0.0446 |

Effectivity indices \(\hat{\theta }_{\text {eff}}\) for Test Problem 1 with \(H_1=\mathbb {Q}_1(h)\) and four choices of \(H_2\), for varying *h* (with *p* fixed) and varying *p* (with *h* fixed)

| \(||u_\mathrm{{ref}}-u_X||_B\) | \(\mathbb {Q}_{2}(h)\) | \(\mathbb {Q}_{4}(h)\) | \(\mathbb {Q}_{1}(h/2)\) | \(\mathbb {Q}_{2}(h/2)\) |
---|---|---|---|---|---|

\(2^{-2}\) | \(1.9254\times 10^{-2}\) | 0.97 | 0.16 | 0.87 | 1.04 |

\(2^{-3}\) | \(1.0244\times 10^{-2}\) | 0.93 | 0.22 | 0.84 | 0.99 |

\(2^{-4}\) | \(6.2277\times 10^{-3}\) | 0.80 | 0.31 | 0.73 | 0.85 |

\(2^{-5}\) | \(4.7187\times 10^{-3}\) | 0.62 | 0.39 | 0.59 | 0.65 |

| \(||u_\mathrm{{ref}}-u_X||_B\) | \(\mathbb {Q}_{2}(h)\) | \(\mathbb {Q}_{4}(h)\) | \(\mathbb {Q}_{1}(h/2)\) | \(\mathbb {Q}_{2}(h/2)\) |
---|---|---|---|---|---|

\(2,\ldots ,6\) | \(\approx 1.02\times 10^{-2}\) | 0.93 | 0.22 | 0.84 | 0.99 |

## 6 Numerical Results

We now return to (1a)–(1b) and assess the quality of the energy error estimator \(\eta \) in (37), extending the discussion in [8] and [10]. First, we select \(X= H_1\otimes P\) and compute \(u_X\in X\) by solving (9). We choose either \(H_1=\mathbb {Q}_1(h)\) or \(H_1=\mathbb {Q}_2(h)\) on a uniform square partition of *D* and fix *P* to be the space of global polynomials with total degree less than or equal to *p* in \(y_1,y_2,\dots , y_M\). Each parameter \(y_{m}\) is assumed to be the image of a mean zero uniform random variable. Hence, for a given multi-index \(\mu \in J_P\) we construct the basis functions in (30) by tensorizing univariate Legendre polynomials. Next, we compute \(\eta = \eta (u_X)\) in (37) by solving (35) and (36), choosing \(H_2\) and *Q* so that the conditions in (31) are satisfied. For \(H_{2}\), we consider the spaces from Examples 1 and 2. We choose *Q* to be the space of polynomials associated with \(J_Q:= \hat{J}_{Q}\backslash J_P\) where \(\hat{J}_Q\) is the set of multi-indices associated with the space of polynomials with total degree less than or equal to \(p+1\) in \(y_1,y_2,\dots , y_M, y_{M+1}\).

*effectivity index*\(\theta _{\text {eff}}:= \eta (u_X)/||u-u_X||_B\) satisfies

Effectivity indices \(\hat{\theta }_\mathrm{{eff}}\) for Test Problem 1 with \(H_1=\mathbb {Q}_2(h)\) and four choices of \(H_2\), for varying *h* (with *p* fixed) and varying *p* (with *h* fixed)

| \(||u_\mathrm{{ref}}-u_X||_B\) | \(\mathbb {Q}_{4}(h)\) | \(\mathbb {Q}_{2}(h/2)\) | \(\mathbb {Q}_{4}^r(h) \) | \(\mathbb {Q}_{2}^r(h/2) \) |
---|---|---|---|---|---|

\(2^{-1}\) | \(4.1729\times 10^{-3}\) | 0.48 | 0.49 | 0.46 | 0.57 |

\(2^{-2,-3,-4}\) | \(\approx 4.09\times 10^{-3}\) | 0.44 | 0.44 | 0.44 | 0.44 |

| \(||u_\mathrm{{ref}}-u_X||_B\) | \(\mathbb {Q}_{4}(h)\) | \(\mathbb {Q}_{2}(h/2)\) | \(\mathbb {Q}_{4}^r(h) \) | \(\mathbb {Q}_{2}^r(h/2) \) |
---|---|---|---|---|---|

2 | \( 4.1134\times 10^{-3}\) | 0.45 | 0.45 | 0.45 | 0.45 |

\(3,\ldots , 6\) | \(\approx 4.10\times 10^{-3}\) | 0.44 | 0.44 | 0.44 | 0.44 |

Since \(||u-u_X||_B\) cannot be computed, we examine \({\hat{\theta }_{\text {eff}} := \eta (u_X)/||u_{\text {ref}}-u_X||_B}\), where \(u_{\text {ref}}\in X_{\text {ref}}\) is a surrogate solution obtained by solving (9) over a sufficiently rich subspace \(X_{\text {ref}}\subset V\) where \(X_{\text {ref}}\supset X\). We define \(X_\text {ref}\) in the same way as *X*, with \(M_{\text {ref}}=10\), \(h_{\text {ref}} = 2^{-7}\) and \(p_{\text {ref}}= 8\). Fixing \(H_1\), *P* and *Q*, we investigate which choice of \(H_2\) consistently leads to \(\hat{\theta }_{\text {eff}}\approx 1\).

### 6.1 Test Problem 1

*M*terms, so that the problem is posed on \(D\times \bar{\varGamma }\), where \(\bar{\varGamma } = \prod _{m=1}^M\varGamma _m\). In that case, \(u_{X}\) and \(u_{\text {ref}}\) are both functions of

*M*parameters. Here,

*u*depends on infinitely many parameters and \(u_{\text {ref}}\) is a function of \(M_{\text {ref}} > M\) parameters.

Effectivity indices \(\hat{\theta }_{\text {eff}}\) for Test Problem 1 with \(H_1=\mathbb {Q}_2(h)\) and four choices of \(H_2\), for varying *h* (with *p* fixed) and varying *p* (with *h* fixed). Modified choice of *Q*

| \(||u_{\text {ref}}-u_X||_B\) | \(\mathbb {Q}_{4}(h)\) | \(\mathbb {Q}_{2}(h/2)\) | \(\mathbb {Q}_{4}^r(h) \) | \(\mathbb {Q}_{2}^r(h/2) \) |
---|---|---|---|---|---|

\(2^{-1}\) | \(4.1729\times 10^{-3}\) | 0.83 | 0.83 | 0.81 | 0.82 |

\(2^{-2,-3,-4}\) | \(\approx 4.09\times 10^{-3}\) | 0.81 | 0.82 | 0.81 | 0.81 |

| \(||u_{\text {ref}}-u_X||_B\) | \(\mathbb {Q}_{4}(h)\) | \(\mathbb {Q}_{2}(h/2)\) | \(\mathbb {Q}_{4}^r(h) \) | \(\mathbb {Q}_{2}^r(h/2) \) |
---|---|---|---|---|---|

2 | \( 4.1191\times 10^{-3}\) | 0.82 | 0.82 | 0.82 | 0.82 |

\(3,\dots , 6\) | \(\approx 4.10\times 10^{-3}\) | 0.81 | 0.81 | 0.81 | 0.81 |

In our first experiment we choose \(H_1 = \mathbb {Q}_1(h)\) and fix \(M=5\) in the definition of *P*. In Table 4 we record \(\hat{\theta }_{\text {eff}}\) for varying *h* with fixed \(p=4\), and varying *p* with fixed \(h=2^{-3}\). We see that \(H_{2}=\mathbb {Q}_2(h/2)\) yields the best error estimator. Interestingly, \(H_{2}=\mathbb {Q}_4(h)\) defines the worst estimator, despite the fact that its associated CBS constant is the smallest (\(\gamma _{\min }^2 \le 0.0121\)). Recall from Theorem 7 that \(||e_{Y_1}||_{B_0} \) and \(||e_{Y_2}||_{B_0}\) provide estimates of the energy error reductions associated with augmenting \(H_{1}\) with \(H_{2}\), and *P* with *Q*, respectively. When \(H_{2}=\mathbb {Q}_4(h)\), since the CBS constant is small, we know \(||e_{Y_1}||_{B_0}\) is a good estimate. When *both* \(||e_{Y_1}||_{B_0} \) and \(||e_{Y_2}||_{B_0}\) are much smaller than \(||e||_B\) (which is true when we use the stated *Q* and \(H_{2}=\mathbb {Q}_{4}(h)\)), the saturation constant \(\beta \approx 1\). This causes \(\eta \) to be much smaller than \(||e||_B\), resulting in a poor effectivity index.

We now repeat the experiment with \(H_1 = \mathbb {Q}_2(h)\). Note that for a fixed *h*, the spatial error associated with \(u_{X}\) is smaller than for \(H_1 = \mathbb {Q}_1(h).\) Results are presented in Table 5. Now, as we vary both *h* (for *p*=4 fixed) and *p* (for \(h=2^{-3}\) fixed), the error \(||u_{\text {ref}}-u_X||_B\) stagnates. The estimated errors behave the same way, but \(\hat{\theta }_{\text {eff}}\) is not close to one. There is little benefit in computing a new Galerkin solution by augmenting either \(H_{1}\) with \(H_{2}\) (for any of the choices of \(H_{2}\)) or *P* with *Q*. The saturation constant is close to one in all cases. However, if introducing more parameters into the approximation space leads to a smaller saturation constant, a better estimate of the error should be obtained by modifying *Q* to include more parameters.

We fix *P* as before with \(M=5\) but now choose *Q* to be the space of polynomials associated with \(J_Q:= \hat{J}_{Q}\backslash J_P\) where \(\hat{J}_Q\) is the set of multi-indices associated with polynomials with total degree less than or equal to \(p+1\) in the first \(M+3\) parameters. Results are presented in Table 6. The effectivity indices are much improved. It is well known that the eigenvalues \(\lambda _m\) associated with (78) decay very slowly (\(\sqrt{\lambda _{m}} = O(m^{-1})\), see [24]). To achieve a small saturation constant, and hence an accurate error estimator, a large number of parameters need to be incorporated into *Q*. We now study a problem with faster decaying coefficients.

### 6.2 Test Problem 2

Effectivity indices \(\hat{\theta }_\mathrm{{eff}}\) for Test Problem 2 with \(H_1=\mathbb {Q}_1(h)\) and four choices of \(H_2\), for varying *h* (with *p* fixed) and varying *p* (with *h* fixed)

| \(||u_\mathrm{{ref}}-u_X||_B\) | \(\mathbb {Q}_{2}(h)\) | \(\mathbb {Q}_{4}(h)\) | \(\mathbb {Q}_{1}(h/2) \) | \(\mathbb {Q}_{2}(h/2) \) |
---|---|---|---|---|---|

\(2^{-3}\) | \(3.0684\times 10^{-2}\) | 0.95 | 0.13 | 1.31 | 0.93 |

\(2^{-4}\) | \(1.5396\times 10^{-2}\) | 0.95 | 0.14 | 1.32 | 0.94 |

\(2^{-5}\) | \(7.7745\times 10^{-3}\) | 0.95 | 0.18 | 1.32 | 0.93 |

| \(||u_{\text {ref}}-u_X||_B\) | \(\mathbb {Q}_{2}(h)\) | \(\mathbb {Q}_{4}(h)\) | \(\mathbb {Q}_{1}(h/2) \) | \(\mathbb {Q}_{2}(h/2) \) |
---|---|---|---|---|---|

2 | \(3.0723\times 10^{-2}\) | 0.95 | 0.13 | 1.31 | 0.93 |

\(3,\dots , 6\) | \(\approx 3.09\times 10^{-2}\) | 0.95 | 0.12 | 1.31 | 0.93 |

Effectivity indices \(\hat{\theta }_\mathrm{{eff}}\) for Test Problem 2 with \(H_1=\mathbb {Q}_2(h)\) and four choices of \(H_2\), for varying *h* (with *p* fixed) and varying *p* (with *h* fixed). \(M=5\) in the definition of *P*

| \(||u_\mathrm{{ref}}-u_X||_B\) | \(\mathbb {Q}_{4}(h)\) | \(\mathbb {Q}_{2}(h/2)\) | \(\mathbb {Q}_{4}^r(h) \) | \(\mathbb {Q}_{2}^r(h/2) \) |
---|---|---|---|---|---|

\(2^{-3}\) | \(2.4871\times 10^{-3}\) | 1.10 | 1.17 | 0.82 | 0.95 |

\(2^{-4}\) | \(1.4017\times 10^{-3}\) | 0.82 | 0.85 | 0.73 | 0.77 |

\(2^{-5}\) | \(1.2813\times 10^{-3}\) | 0.71 | 0.71 | 0.70 | 0.70 |

| \(||u_\mathrm{{ref}}-u_X||_B\) | \(\mathbb {Q}_{4}(h)\) | \(\mathbb {Q}_{2}(h/2)\) | \(\mathbb {Q}_{4}^r(h) \) | \(\mathbb {Q}_{2}^r(h/2) \) |
---|---|---|---|---|---|

2 | \(3.1896\times 10^{-3}\) | 1.03 | 1.08 | 0.86 | 0.93 |

3 | \(2.5511\times 10^{-3}\) | 1.10 | 1.16 | 0.83 | 0.95 |

4, 5, 6 | \(\approx 2.49\times 10^{-3}\) | 1.10 | 1.18 | 0.82 | 0.95 |

## 7 Summary and Conclusions

Using classical theory from [1, 6] for Galerkin approximation, we provided an alternative derivation of an error estimator from [10] and the associated bound. Our approach highlights the straightforward extension of an error estimation strategy for standard Galerkin FEMs for deterministic PDEs to SGFEMs for parameter-dependent PDEs. The quality of the estimator depends on a CBS constant associated with two finite element spaces \(H_1\) and \(H_2\). For \(H_1 = \mathbb {Q}_1(h)\) and \(H_{1}=\mathbb {Q}_2(h)\) we investigated non-standard choices of \(H_2\) which lead to small CBS constants. When \(H_1 = \mathbb {Q}_1(h)\) and \(H_2\) satisfies certain conditions, we derived new theoretical estimates for the CBS constant. In Sect. 6 we demonstrated that the best choice of \(H_2\) for constructing an effective error estimator is not necessarily the space that leads to the smallest CBS constant. Through numerical experiments, we demonstrated that *Q* must also be carefully selected and tailored to properties of the diffusion coefficient. When both \(H_2\) and *Q* are chosen appropriately, the estimator exhibits effectivity indices close to one.

Choosing \(H_{2}\) and *Q* so that the effectivity index is close to one is not the end of the story. If the estimated error associated with \(u_{X} \in X=H_{1} \otimes P\) is too high, we need to decide how to enrich *X* and compute a new approximation. The error estimate needs to be accurate, but to derive adaptive algorithms using (44)–(45), we should only work with spaces \(H_2\) and *Q* such that it is straight-forward to compute new SGFEM approximations in \((H_{1}\oplus H_{2}) \otimes P\) and/or \(H_{1} \otimes (P \oplus Q)\). For example, when \(H_1 = \mathbb {Q}_1(h)\), choosing \(H_2=\mathbb {Q}_2(h)\) yields an accurate error estimate for the current approximation, but does not give a feasible spatial adaptive enrichment strategy. Choosing \(H_{2}=\mathbb {Q}_1(h/2)\) is more natural. Fortunately, this space also yields a good error estimator. When \(H_1 = \mathbb {Q}_2(h)\), \(H_{2}=\mathbb {Q}_2^r(h/2)\) yields an excellent estimator. Although using \(H_{2}=\mathbb {Q}_2(h/2)\) is more natural for adaptivity, we recommend using \(H_{2} =\mathbb {Q}_2^r(h/2)\) to estimate the error. Not only is this cheapest option of all those considered, since \(\mathbb {Q}_2(h/2)\) is richer, the estimated spatial error reduction \(||e_{Y_1}||_{B_0}\) obtained using \(H_{2}=\mathbb {Q}_2^r(h/2)\) is still informative, if we wish to assess the benefit of computing a new approximation in \(\left( \mathbb {Q}_2(h)\oplus \mathbb {Q}_2(h/2)\right) \otimes P\).

## Notes

### Acknowledgements

We thank David Silvester and Alexei Bespalov for valuable discussions and contributions to the MATLAB toolbox S-IFISS [7], which we used to produce our numerical results.

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