Goal-Oriented Error Estimation for the Automatic Variationally Stable FE Method for Convection-Dominated Diffusion Problems

1 Introduction

In adaptive mesh refinement algorithms, a posteriori error estimation Ainsworth and Oden (2011); Oden et al. (1989) is needed to provide quantified assessments of the numerical approximation error, as well as error indicators to guide the adaptive process. Residual-based goal-oriented error estimates have been developed for multiple applications and FE methods, e.g, see Oden and Prudhomme (2002); Becker and Rannacher (2001); Giles and Süli (2002); Ladevèze et al. (2013); Prudhomme and Oden (1999). It involves the solution of a dual problem, which for Bubnov-Galerkin and Petrov-Galerkin methods Bubnov (1913); Petrov (1940); Hughes (2012); Oden and Reddy (2012); Reddy (1993) suffers from the same numerical instabilities as the primal problem in the presence of convection. Thus, the FE meshes for the dual problem have to be refined so as to adequately capture any boundary or internal layers and thereby avoid any numerical instabilities. It makes the classical methods unsuitable for goal-oriented error estimation for this class of problems.

However, goal-oriented error estimates have been successfully applied to conditionally stable FE methods by several authors for convection-diffusion problems Kuzmin and Korotov (2010); Cnossen et al. (2006); Schwegler et al. (2016); Formaggia et al. (2001); Schwegler (2014). In these works, stabilization schemes such as the streamlined-upwind Petrov-Galerkin (SUPG) Brooks and Hughes (1982) method are used to stabilize both the primal and dual problems. The reported effectivity of the estimates varies based on the stabilization chosen, type of error to be estimated (approximation vs. modeling), and the Quantity of interest (QoI) used. In Schwegler (2014); Schwegler et al. (2016), Schwegler et al. explicitly investigate the stabilization of the dual problem and its influence on the estimate. Stabilized discontinuous Galerkin (DG) methods have also been applied to goal-oriented error estimates for convection-diffusion problems with success, we refer to Mozolevski and Prudhomme (2015) and references therein. While these error estimation efforts have been successful, the conditionally stable nature of the methods do require a priori analyses to properly establish the parameters needed to achieve stability which can be extremely arduous and have to be done on a problem-by-problem basis.

Unconditionally stable FE methods such as the least squares FE method (LSFEM) Bochev and Gunzburger (2009) and the DPG method Demkowicz and Gopalakrishnan (2010, 2015); Carstensen et al. (2014); Demkowicz and Gopalakrishnan (2011b, a, 2012) resolve the issue of conditional numerical instability. However, these methods generally use a built-in a posteriori error estimator based on the error in the energy norm induced by its bilinear form to drive adaptivity (see, e.g., Carstensen et al. (2014); Niemi et al. (2013); Demkowicz and Gopalakrishnan (2012); Keith et al. (2016)). The simplicity and quality of this type of estimator make it the most commonly employed for residual minimization techniques such as the DPG method and LSFEM. In Keith et al. (2017), Keith et al. introduce the concept of goal-oriented adaptive mesh refinement for the DPG method. However, their goal is not so much to estimate the errors in FE computations but rather the introduction of a new duality theory that is used as a vessel for new adaptive mesh refinement strategies. In Chaudhry et al. (2014), the least squares functional is modified by adding terms incorporating QoIs to enhance the quality of the built-in estimator used to drive the adaptive mesh refinement. A similar approach is taken by Cai, Ku et al. in Cai and Ku (2011); Ku and Park (2010); Ku (2011) to both estimate errors and drive mesh adaptive algorithms.

While the error estimation in the aforementioned references has been successful, the stabilization efforts required can be arduous by demanding in-depth a priori error analyses on a problem-by-problem basis. In addition, to our best knowledge there are no published results for the DPG method that state the effectivity of goal-oriented a posteriori error estimates for convection-diffusion problems. Goal-oriented error estimation in the LSFEM for convection-dominated diffusion problems is less attractive due to the highly diffusive nature of its FE approximations for coarse meshes which can lead to estimates with poor accuracy. Our goal is therefore to introduce a new framework for the goal-oriented a posteriori error estimation for the automatically stable AVS-FE method that delivers highly accurate predictions of the error in user defined QoIs.

The AVS-FE method introduced by Calo, Romkes and Valseth Calo et al. (2018) provides a functional setting to analyze singularly perturbed problems, such as convection-dominated diffusion. The AVS-FE method is a hybrid between the DPG method Demkowicz and Gopalakrishnan (2010, 2015); Carstensen et al. (2014); Demkowicz and Gopalakrishnan (2011b, a, 2012) and the classical mixed FE methods in the sense that the trial space consists of globally continuous functions, while the test space consists of piecewise discontinuous functions. Attractive features of the AVS-FE method are its unconditional numerical stability property (regardless of the underlying differential operator), its highly accurate flux approximations, and the ability to compute optimal test functions element-by-element.

In the following, we limit our focus to stationary scalar-valued convection-diffusion problems. In Section 2, we introduce the model problem of 2D scalar-valued convection-dominated diffusion, used notations, as well as a review of the AVS-FE methodology. Goal-oriented a posteriori error estimates are introduced in Sections 3 and 4. In Section 3, we introduce the goal-oriented error estimates for the AVS-FE method following the classical approach of Becker and Rannacher Becker and Rannacher (2001), i.e., the dual solution is sought in the (broken) test space, and present a numerical verification for the Laplace BVP. In Section 4, we present a new alternative approach to goal-oriented error estimation in which we seek $C^{0}$ or Raviart-Thomas AVS-FE approximations of the dual solution by using the underlying dual BVP. Numerical verifications investigating the effectivity and robustness of the new estimator are also presented in Section 4. Goal-oriented adaptive mesh refinements and numerical verifications are presented in Section 5. Lastly, conclusions and future work are discussed in Section 6.

2 Variationally Stable Analysis for Finite Element Computations

In this section, we introduce our convection-diffusion model problem and briefly present a review of the AVS-FE method. A more detailed introduction can be found in Calo et al. (2018); Valseth (2019).

2.1 Model Problem and Notation

Let $Ω \subset R^{2}$ be an open bounded domain with Lipschitz boundary $\partial Ω$

and outward unit normal vector

n

. The boundary

\partial Ω

consists of open subsections

Γ_{D}, Γ_{N} \subset \partial Ω

, such that

Γ_{D} \cap Γ_{N} = \emptyset

and

\partial Ω = ¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ Γ_{D} \cup Γ_{N}

. For our model problem, we consider the following linear convection-diffusion PDE in

Ω

with homogeneous Dirichlet conditions on

Γ_{D}

and (possibly) non-homogeneous Neumann conditions on

Γ_{N}

\begin{matrix} Find u such that: \begin{matrix} - \nabla \cdot (D \nabla u) + b \cdot \nabla u & = & f, in Ω, u & = & 0, on Γ_{D}, D \nabla u \cdot n & = & g, on Γ_{N}, \end{matrix} \end{matrix}

(1)

where $D$

denotes the second order diffusion tensor, with symmetric and elliptic coefficients

D_{i j} \in L^{\infty} (Ω)

;

b \in [L^{\infty} (Ω)]^{2}

the convection coefficient;

f \in L^{2} (Ω)

the source function; and

g \in H^{- 1 / 2} (Γ_{N})

the Neumann data.

2.2 The AVS-FE Weak Formulation

For the sake of brevity, we only mention the few key points here of the derivation of a weak formulation for the AVS-FE method. We refer to Calo et al. (2018); Valseth (2019) for a more detailed treatment. We start by introducing a regular partition $P_{h}$ of $Ω$ into elements $K_{m}$ , such that:

Ω = int (⋃ K_{m} \in P_{h}_{m}), K_{m} \cap K_{n}, m \neq n .

The partition $P_{h}$ is such that any discontinuities in $D_{i j}$ or $b$ are restricted to the boundaries of each element $\partial K_{m}$ . We introduce an auxiliary flux variable $q = {q_{x}, q_{y}}^{T} = D \nabla u$ , and recast (1) as a system of first-order PDEs:

\begin{matrix} Find (u, q) \in H^{1} (Ω) \times H (div, Ω) such that: \begin{matrix} D \nabla u - q & = & 0, in Ω, - \nabla \cdot q + b \cdot \nabla u & = & f, in Ω, u & = & 0, on Γ_{D}, q \cdot n & = & g, on Γ_{N} . \end{matrix} \end{matrix}

(2)

By weakly enforcing the system of PDEs (2) locally on each element $K_{m} \in P_{h}$ , applying Green’s identity to the term including the divergence of $q$ , applying Dirichlet and Neumann conditions on $\partial K_{m} \cap Γ_{D}$ and $\partial K_{m} \cap Γ_{N}$ , respectively, and subsequently summing all the local contributions we arrive at the following equivalent global variational formulation:

(3)

Here, the bilinear form, $B : U (Ω) \times V (P_{h}) ⟶ R$ , and linear functional, $F : V (P_{h}) ⟶ R$ , are defined as follows:

(4)

where the trial and test function spaces, $U (Ω)$ and $V (P_{h})$ , are:

\begin{matrix} U (Ω) def = {(u, q) \in H^{1} (Ω) \times H (div, Ω) : γ_{0}^{m} (u)_{| Γ_{D}} = 0}, V (P_{h}) def = {(v, w) \in H^{1} (P_{h}) \times [L^{2} (Ω)]^{2} : γ_{0}^{m} (v_{m})_{| \partial K_{m} \cap Γ_{D}} = 0, \forall K_{m} \in P_{h}}, \end{matrix}

(5)

with norms $∥ \cdot ∥_{U (Ω)} : U (Ω) ⟶ [0, \infty)$ and $∥ \cdot ∥_{V (P_{h})} : V (P_{h}) ⟶ [0, \infty)$ :

(6)

where $h_{m} = diam (K_{m})$ . The operators $γ_{0}^{m} : H^{1} (K_{m}) :⟶ H^{1 / 2} (\partial K_{m})$ and $γ_{n}^{m} : H (div, K_{m}) ⟶ H^{- 1 / 2} (\partial K_{m})$ denote the local trace and normal trace operators (e.g., see Girault and Raviart (1986)); and $n_{m}$ is the outward unit normal vector to the element boundary $\partial K_{m}$ . Note that we employ an engineering notation convention here by using an integral representation of the boundary integrals rather than that of a duality pairing. The variational formulation (3) is essentially a DPG formulation in which only the space $V (P_{h})$ is broken.

Lemma 2.0

Let $f \in (H^{1} (P_{h}))^{'}$ and $g \in H^{- 1 / 2} (Γ_{N})$ . Then, the weak formulation (3) is well posed and has a unique solution.

Proof: The kernel of $B (\cdot, \cdot)$ is trivial, which implies that the following defines a norm on $U (Ω)$ , called the energy norm:

∥ (u, q) ∥_{B} def = sup (v, w) \in V (P_{h}) ∖ {(0, 0)} \frac{| B ((u, q); (v, w)) |}{∥ (v, w) ∥_{V (P_{h})}} .

(7)

Per the DPG philosophy, $B (\cdot, \cdot)$ then inherently satisfies the continuity and Inf-Sup conditions of the Lax-Milgram Theorem in terms of the energy norm $∥ (u, q) ∥_{B}$ , which concludes the proof. A detailed proof can be found in Valseth (2019).
∎

Remark 2.0

It is possible to derive other variational statements in which the trial space is continuous and the test space is discontinuous. These will be considered in a forthcoming paper.

2.3 AVS-FE Discretization

The AVS-FE method seeks numerical approximations $(u^{h}, q^{h})$ of $(u, q)$ of the variational formulation (3) by using classical FE bases for the trial functions $(u^{h}, q^{h})$ , i.e., we represent the approximations as linear combinations of the trial basis functions $(e^{i} (x), (E_{x}^{j} (x), E_{y}^{k} (x))) \in U^{h} (Ω)$

and their corresponding degrees of freedom:

u^{h} (x) = N \sum i = 1 u_{i}^{h} e^{i} (x), q_{x}^{h} (x) = N \sum j = 1 q_{x}^{h, j} E_{x}^{j} (x), q_{y}^{h} (x) = N \sum k = 1 q_{y}^{h, k} E_{y}^{k} (x) .

(8)

Since the solution space $U (Ω)$ concerns $H^{1} (Ω)$ and $H (div, Ω)$ spaces, the FE discretizations can employ classical $C^{0} (Ω)$ or Raviart-Thomas functions.

The test space $V (P_{h})$ , however, is discontinuous, allowing us to construct piecewise discontinuous optimal test functions that yield unconditionally stable discretizations. These functions are constructed by employing the DPG philosophy Demkowicz and Gopalakrishnan (2010, 2015); Carstensen et al. (2014); Demkowicz and Gopalakrishnan (2011b, a, 2012) in which optimal test functions are defined by global weak problems. Thus, for the trial functions $e^{i} (x), E_{x}^{j} (x), and E_{y}^{k} (x)$ , the corresponding global optimal test functions $({~ e}^{i}, {~ E}^{i})$ , $({~ e}_{x}^{j}, {~ E}_{x}^{j})$ , and $({~ e}_{y}^{k}, {~ E}_{y}^{k})$ are the solutions of the following Riesz representation problems Valseth (2019); Calo et al. (2018), respectively:

(9)

where ${((\cdot, \cdot); (\cdot, \cdot))}_{V (P_{h})}$ denotes the broken inner product on $V (P_{h})$ , defined by:

{((r, z); (v, w))}_{V (P_{h})} def = \sum K_{m} \in P_{h} \int_{K_{m}} [h_{m}^{2} \nabla r_{m} \cdot \nabla v_{m} + r_{m} v_{m} + z_{m} \cdot w_{m}] d x .

(10)

Remark 2.0

Remarkably, the broken nature of the test space $V (P_{h})$ allows us to compute numerical approximations of the local restrictions of the optimal test functions in a completely decoupled fashion (see Valseth (2019); Calo et al. (2018) for details). Thus, we solve local restrictions of (9), e.g.,

\begin{matrix} {((r, z); ({~ e}_{h}^{i}, {~ E}_{h}^{i}))}_{V (K_{m})} & = & B_{| K_{m}} ((e^{i}, 0); (r, z)), & \forall (r, z) \in V (K_{m}), {((r, z); ({~ e}_{x_{h}}^{j}, {~ E}_{x_{h}}^{j}))}_{V (K_{m})} & = & B_{| K_{m}} ((0, (E_{x}^{j}, 0)); (r, z)), & \forall (r, z) \in V (K_{m}), {((r, z); ({~ e}_{y_{h}}^{k}, {~ E}_{y_{h}}^{k}))}_{V (K_{m})} & = & B_{| K_{m}} ((0, (0, E_{y}^{k})); (r, z)), & \forall (r, z) \in V (K_{m}), \end{matrix}

(11)

where $B_{| K_{m}} (\cdot; \cdot)$ denotes the restriction of $B (\cdot; \cdot)$ to the element $K_{m}$ . Hence, while the optimal test functions are defined by global weak statements, their numerical computation can be performed element-by-element.

Remark 2.0

Numerical verifications reveal that the local test functions can be computed by using a degree of approximation that is identical to the degree of approximation of their corresponding trial functions.

Remark 2.0

The choice of $C^{0}$ or Raviart-Thomas trial functions has the consequence that the optimal test functions have the same support as the trial functions Valseth (2019); Calo et al. (2018).

Finally, we introduce the FE discretization of (3) governing the AVS-FE approximation $(u^{h}, q^{h}) \in U (Ω)$ of $(u, q)$ :

\begin{matrix} Find & (u^{h}, q^{h}) \in U^{h} (Ω) such that: B ((u^{h}, q^{h}); (v^{*}, w^{*})) = F (v^{*}), \forall (v^{*}, w^{*}) \in V^{*} (P_{h}), \end{matrix}

(12)

where the finite dimensional subspace of test functions $V^{*} (P_{h}) \subset V (P_{h})$ is spanned by the numerical approximations of the test functions ${({~ e}_{h}^{i}, {~ E}_{h}^{i})}_{i = 1}^{N}$ , ${({~ e}_{x_{h}}^{j}, {~ E}_{x_{h}}^{j})}_{j = 1}^{N}$ , and ${({~ e}_{y_{h}}^{k}, {~ E}_{y_{h}}^{k})}_{k = 1}^{N}$ , as computed from the test function problems (9) and (11).

Since we use the DPG methodology here to construct the optimal test space $V^{*} (P_{h})$ , the discrete problem (12) inherits the continuity and inf-sup constants of the continuous problem (3) (see Valseth (2019)). It is therefore unconditionally stable for any choice of mesh parameters $h_{m}$ and $p_{m}$ . The corresponding global stiffness matrices are symmetric and positive definite.

In the following sections we derive error estimates in terms of user defined QoIs of the solution. The QoIs are represented in terms of continuous linear functionals $Q_{i} : U (Ω) \to R, i = 1, 2, \dots, N_{Q}$ , for example:

Q_{i} (u, q) = \frac{1}{| ω |} \int_{ω} u d x,

(13)

Thus, the goal is to estimate the error $Q (u - u^{h}, q - q^{h})$ . We introduce residual based a posteriori estimates by taking two distinctive approaches to the solution of the dual problem. The first follows the approach introduced by Becker and Rannacher Becker and Rannacher (2001) and therefore seeks a dual solution in the broken primal test space $V (P_{h})$ . The second approach concerns an alternative approach in which the AVS-FE solution of the underlying strong form of the dual problem is sought in a $H^{1} (Ω) \times H (div, Ω)$ subspace of the primal test space.

3 Goal-Oriented Error Estimation - Classical Approach

Following Becker and Rannacher Oden and Prudhomme (2002); Becker and Rannacher (2001); Prudhomme and Oden (1999) we state the following classical lemma of goal-oriented error estimation:

Lemma 3.0

Let $(u, q)$ be the exact solution of the first-order system (2), $(u^{h}, q^{h})$ the AVS-FE approximation of $(u, q)$ per (12), and $(p_{i}, r_{i}) \in V (P_{h})$ a dual solution for each QoI, governed by:

\begin{matrix} Find (p_{i}, r_{i}) \in V (P_{h}) & such that: B ((v, w); (p_{i}, r_{i})) = Q_{i} (v, w), \forall (v, w) \in U (Ω) . \end{matrix}

(14)

Then, the error in the QoI $E_{i} (u^{h}, q^{h}) = Q_{i} (u - u^{h}, q - q^{h})$ , is governed by the identity:

E_{i} (u^{h}, q^{h}) def = R_{h} ((u^{h}, q^{h}); (p_{i}, r_{i})),

(15)

where $R_{h} ((\cdot, \cdot); (\cdot, \cdot))$ is the residual functional:

R_{h} ((u, q); (v, w)) = F (v) - B ((u, q); (v, w)) .

(16)

Remark 3.0

Analogous to the well-posedness of the primal problem (3) (see Lemma 2.1), the dual problems (14) are well posed (since the kernel of the adjoint operator of $B (\cdot, \cdot)$ is also trivial).

To compute estimates of the error $E_{i} (u^{h}, q^{h})$ through (15), we compute approximations of the dual solutions $(p_{i}^{h}, r_{i}^{h})$ by following the classical approach of Oden and Prudhomme (2002); Becker and Rannacher (2001); Prudhomme and Oden (2003). Thus, for a given QoI, the approximate dual solution $(p^{h}, r^{h})$ is governed by:

\begin{matrix} Find & (p^{h}, r^{h}) \in V^{*} (P_{h}) such that: B ((v^{h}, w^{h}); (p^{h}, r^{h})) = Q (v^{h}, w^{h}), \forall (v^{h}, w^{h}) \in U^{h} (Ω) . \end{matrix}

(17)

We seek $(p^{h}, r^{h})$ in the discrete, broken space $V^{*} (P_{h})$ spanned by $({~ e}_{h}^{i}, {~ E}_{h}^{i})$ of the local Riesz representation problems (11). Hence, we use the same element partition $P_{h}$ of $Ω$ as we used for the primal problem to compute $(p^{h}, r^{h})$ . However, due to the Galerkin orthogonality condition of the numerical approximation error, the approximate dual solution has to be sought by using polynomial approximations that are of higher order than the trial functions used to solve the discrete primal problem (12). We choose $p + 1$ . Hence, the unconditional numerical stability of the AVS-FE methodology will allow the computation of approximate dual solutions $(p_{i}^{h}, r_{i}^{h})$ for any choice of mesh parameters $h_{m}$ and $p_{m}$ . The estimated error $η_{e s t}$ in the quantity of interest is then computed by:

η_{e s t} \approx E_{i}^{h} (u^{h}, q^{h}) = R_{h} ((u^{h}, q^{h}); (p_{i}^{h}, r_{i}^{h}))

(18)

This classical approach has been shown to be very successful in a wide range of applications, especially those those in which the differential operator is self-adjoint (e.g., see Oden and Prudhomme (2002); Prudhomme and Oden (2003)).

As a numerical verification of this estimator, we consider the Laplace problem on the unit square $Ω = (0, 1) \times (0, 1) \subset R^{2}$ with homogeneous Dirichlet boundary conditions:

\begin{matrix} - Δ u = f, & in Ω, u = 0, & on \partial Ω . \end{matrix}

(19)

The source function $f$ , is chosen such that the exact solution is given by:

u (x, y) = e^{[50 (x^{2} - x) (y^{2} - y)]} - 1.

The QoI is chosen to be the average of the solution $u$ in the region $ω = (0.5, 1) \times (0.5, 1) \subset Ω$ :

Q (u, q) = \frac{1}{| ω |} \int_{ω} u d x .

(20)

To estimate the error in this QoI (20) we apply the AVS-FE discretization to the primal and dual problem with polynomial degrees of approximation of 2 and 3, respectively. As in Calo et al. (2018), we use $C^{0}$ continuous bases for both trial variables $u^{h}$ and $q^{h}$ of the same polynomial degree, while for $p^{h}$ and $r^{h}$ we use the optimal bases determined by the Riesz representation problems (9).

Primal dofs	$Q (u, q) - Q (u^{h}, q^{h})$	Dual dofs	$η_{e s t}$	$I_{e f f}$
27	-9.2601e+00	48	-1.0601e+01	1.145
75	2.3191e-02	147	-7.5337e-02	-0.325
243	1.8610e-02	507	3.4581e-02	1.858
867	-3.7844e-05	1875	-3.8616e-04	10.204
3267	-2.5778e-05	7203	-5.3075e-05	2.059

Table 1: Error estimation results for the Laplace problem with QoI (20) using the classical approach, i.e., through (17) and (18).

(a) Dual solution in the classical approach.

To assess the quality of the error estimate, we introduce the effectivity index:

I_{e f f} = \frac{η_{e s t}}{Q (u, q) - Q (u^{h}, q^{h})} .

(21)

In Table 1, we present the error estimates for increasingly refined meshes. It appears that the magnitude of the estimated error decreases monotonically. However, the results for the effectivity index $I_{e f f}$ reveal that the estimates generally have poor accuracy nor exhibit any consistent evolution during the uniform $h -$ refinements. We suspect that since the continuity of the dual solution is enforced weakly in (14), it prohibits from adequately resolving the discretization of the dual solution on the used meshes. Comparison of the approximate solution $p^{h}$ in Figure 1 to an overkill approximation of $p$ in Figure 1(b), reveals internal oscillations in $p^{h}$ in each element as well as along the global boundaries. While bounded, these contribute to the poor quality of the error estimates. This behavior is persists for numerical verifications in which the degree of approximation for the dual problem is $p_{d u a l} = p_{p r i m a l} + 2, p_{p r i m a l} + 3$ etc.

4 Goal-Oriented Error Estimation - Alternative Approach

Since we suspect that the poor accuracy of the estimator via the classical approach is likely caused by the discontinuous character of the numerical approximation of the dual solutions $(p^{h}, r^{h})$ , we propose an alternative approach for computing $(p^{h}, r^{h})$ . Instead of seeking discontinuous discrete approximations of the dual solution by using the corresponding dual weak formulation (14) of the primal problem (3), we rather reconsider the underlying strong form of each dual problem, i.e.,

(22)

where $θ_{i} \in U^{'} (Ω)$ is such that $⟨ θ_{i}, (u, q) ⟩_{U^{'} \times U} = Q_{i} (u, q)$ . We subsequently derive a weak statement governing $(p_{i}, r_{i})$ by using the same approach as the applied to the derivation of the weak statement of the primal problem (see Section 2). Thus, we seek $p_{i} \in H^{1} (Ω)$ , $r_{i} \in H (div, Ω)$ and employ test spaces for the dual problem that are broken. Hence, $p_{i}$ and $r_{i}$ belong to the same globally (weakly) continuous function spaces as the primal solution $(u, q)$ . To derive the dual weak statement, we follow the derivation in Calo et al. (2018) and enforce the system (22) weakly on each element $K_{m} \in P_{h}$ , apply Green’s Identity, enforce boundary conditions, and arrive at the following weak statement:

\begin{matrix} Find (p_{i}, r_{i}) \in U (Ω) & such that:^B ((v, w); (p_{i}, r_{i})) = Q_{i} (v, w), \forall (v, w) \in W (P_{h}), \end{matrix}

(23)

where:

^B((v,w);(pi,ri))=∑Km∈Ph{∫Km[(∇pi−ri)⋅wm+Dri⋅∇vm−(b⋅∇pi)vm]dx−∮∂Km∖¯¯¯¯¯¯∂Ωγmn(Dri)γm0(vm)ds},

(24)

and:

\begin{matrix} W (P_{h}) def = {(v, w) \in H^{1} (P_{h}) \times [L^{2} (Ω)]^{2} : γ_{0}^{m} (v_{m})_{| \partial K_{m} \cap \partial Ω} = 0, \forall K_{m} \in P_{h}} . \end{matrix}

(25)

It should be noted here that $^B (\cdot; \cdot)$ and $B (\cdot; \cdot)$ differ in the sign in front of the convection vector $b$ due to the non-self adjoint character of the differential operator of (1). Now, to ensure the unconditional stability of the discrete dual problem we use optimal discontinuous test functions $({~ e}^{i}, {~ E}^{i})$ , $({~ e}_{x}^{j}, {~ E}_{x}^{j})$ , and $({~ e}_{y}^{k}, {~ E}_{y}^{k})$ for the dual problem that are solutions of the following (Riesz) weak problems:

\begin{matrix} {((x, z); ({~ e}^{i}, {~ E}^{i}))}_{V (P_{h})}^{i} & = & ^B ((x, z); (e^{i}, 0)), & \forall (x, z) \in V (P_{h}), i = 1, \dots, N, {((x, z); ({~ e}_{x}^{j}, {~ E}_{x}^{j}))}_{V (P_{h})} & = & ^B ((x, z); (0, (E_{x}^{j}, 0))), & \forall (x, z) \in V (P_{h}), j = 1, \dots, N, {((x, z); ({~ e}_{y}^{k}, {~ E}_{y}^{k}))}_{V (P_{h})} & = & ^B ((x, z); (0, (0, E_{y}^{k}))), & \forall (x, z) \in V (P_{h}), k = 1, \dots, N . \end{matrix}

(26)

Remark 4.0

The vector valued dual solution $r_{i}$ belongs to $H (div, Ω)$ due to the boundary integral $\oint_{\partial K_{m} ∖ ¯ ¯¯¯¯ ¯ \partial Ω} γ_{n}^{m} (D r_{i}) γ_{0}^{m} (v_{m}) d s$ in (23). For this integral to be Lebesgue integrable, $D r_{i}$ has to belong to $H^{- 1 / 2} (\partial K_{m})$ which implies $D r_{i} \in H (div, Ω)$ .

We then establish the error estimator ${^η}_{e s t}$ by using the new dual solutions:

(27)

Having established the new alternative error estimates, we propose to employ an error indicator $ϵ_{m}$ corresponding to the restriction of the goal-oriented error estimate ${^η}_{e s t}$ in mesh adaptive refinements, i.e,

ϵ_{m} = R_{h} |_{K_{m}} ((u^{h}, q^{h}); (p_{i}^{h}, r_{i}^{h})) .

(28)

4.1 Numerical Verification - Diffusion Problem

We again solve (19) using identical meshes and degrees of approximation as in Section 3, i.e., quadratic primal and cubic dual approximations, respectively. In this alternative approach, we seek $C^{0}$ continuous solutions to both the primal and dual problems in which the scalar and flux variables are of the same polynomial degree. Again, we assess the quality of the error estimate with the effectivity index:

_{e f f} = \frac{{^η}_{e s t}}{Q (u, q) - Q (u^{h}, q^{h})} .

(29)

Primal dofs	$Q (u, q) - Q (u^{h}, q^{h})$	Dual dofs	${^η}_{e s t}$	$_{e f f}$
243	1.8610e-02	507	1.8602e-02	9.9952e-01
867	-3.7845e-05	1875	-3.6160e-05	9.5547e-01
3267	-2.5778e-05	7203	-2.3479e-05	9.1081e-01

Table 2: Error estimation results for the Laplace problem with QoI (20) using the new alternative approach, i.e., through (23) and (18).

In Table 2, we show the results for the error estimates for uniform mesh refinements. As in Section 3, the magnitude of the error decreases monotonically. However, the effectivity index now shows that the estimates have very good accuracy with values close to unity. Furthermore, comparison of Figure 2 of the dual AVS-FE solution to an overkill solution reveals that there are no oscillations at element interiors and global boundaries.

4.2 Numerical Verifications - Convection-Dominated Diffusion

Next, we consider a more challenging case of a convection-dominated diffusion problem. In this subsection, we consider a simplified form of our model problem (1) on the unit square with homogeneous Dirichlet boundary conditions:

\begin{matrix} - \frac{1}{P e} Δ u + b \cdot \nabla u = f, & in Ω = (0, 1) \times (0, 1), u = 0, & on \partial Ω, \end{matrix}

(30)

where the Peclet number $P e = 100$ is and $b = {1, 1}^{T}$ the convection coefficient. We consider the case of (30) in which the above source function $f$ is chosen such that the exact solution is given by:

u (x, y) = [x + \frac{e^{P e \cdot b_{x} \cdot x} - 1}{1 - e^{P e \cdot b_{x}}}] [y + \frac{e^{P e \cdot b_{y} \cdot y} - 1}{1 - e^{P e \cdot b_{y}}}] .

(31)

Thus, the solution exhibits boundary layers along $x = 1$ and $y = 1$ with a width of $\frac{1}{P e} = 0.01$ , as shown in Figure 3.

Figure 3: Exact solution $u$ of the simplified model problem (30) with $b = {1, 1}^{T}$ and $P e = 100$ .

4.2.1 Uniform Meshes

First, we consider uniform meshes consisting of quadrilateral elements. The first QoI is chosen as in (20), i.e., the average solution $u$ in the top right quadrant of the unit square. In Figure 4, we show the corresponding overkill solution $p$ of the dual problem.

The dual solution, similar to the primal, exhibits boundary layers. However, as the direction of the convection is reversed from the primal problem, the layers are at opposite edges of the domain. To verify the new estimator ${^η}_{e s t}$ we employ AVS-FE discretizations of the primal and dual problem with polynomial degrees of approximation of 2 and 3, respectively. The corresponding numerical results are illustrated in Table 3. The effectivity indices show that the error estimator accurately measures the approximation error.

Primal dofs	$Q (u, q) - Q (u^{h}, q^{h})$	Dual dofs	${^η}_{e s t}$	$_{e f f}$
243	2.8825e-01	507	1.6140e-01	0.599
867	1.7711e-01	1875	1.5010e-01	0.848
3267	6.7393e-02	7203	6.5077e-02	0.966
12675	1.3225e-02	28227	1.3158e-02	0.995
49923	1.3918e-03	111747	1.3909e-03	0.999
198147	1.0321e-04	444675	1.0321e-04	0.999

Table 3: Results for the convection-dominated diffusion problem (30) with QoI (20) for uniform mesh refinements and

P e = 100

For initial, coarse, meshes the effectivity index may not be very close to unity, but is of the same order of magnitude as the exact error. As the uniform meshes are further refined, the estimate converges to the exact error and delivers highly accurate predictions of the error with the effectivity index $_{e f f}$ very close to unity.

The second QoI we consider is the average flux in the $x$ -direction in the lower left quadrant in the unit square, i.e., $ω = (0, 0.5) \times (0, 0.5)$ :

Q (u, q) = \frac{1}{| ω |} \int_{ω} \frac{\partial u}{\partial x} d x .

(32)

The polynomial degrees of approximation are now 1 and 2 for the primal and dual problem, respectively.

Primal dofs	$Q (u, q) - Q (u^{h}, q^{h})$	Dual dofs	${^η}_{e s t}$	$I_{e f f}$
867	3.5614e-01	3267	1.7165e-01	0.482
3267	1.8049e-01	12675	2.1098e-01	1.169
12675	6.2143e-02	49923	9.0932e-02	1.463
49923	1.7254e-02	198147	2.2595e-02	1.390
198147	4.4412e-03	789507	4.9300e-03	1.110

Table 4: Results for the convection-dominated diffusion problem (30), with QoI (32) for uniform mesh refinements and

P e = 100

As shown in Table 4, the errors in a QoI in terms of a derivative are slightly higher than those of the preceding numerical verification which is to be expected (see Calo et al. (2018); Valseth (2019)). While again for coarse meshes the estimate does not accurately assess the error, it does capture the right order of magnitude and improves significantly upon mesh refinements as the effectivity indices $_{e f f}$ approaches unity.

4.2.2 Non-Uniform Mesh

So far, we have only considered rectangular uniform meshes. To provide a more realistic scenario, as encountered in engineering applications, we consider a mesh in which the elements are skewed and the element edges do not align with the direction of the convection (see Figure

5).

We consider the same convection-dominated diffusion PDE (30), but now with $P e = 200$ . The QoI is again the average of the solution $u$ in the region $ω = (0.5, 1) \times (0.5, 1)$ (i.e., see (20)).

In Table 5, we list the results for the case in which the primal degree of approximation is 2, and the dual degree of approximation is 3. After each computation of the primal and dual solutions, all elements in the mesh are uniformly refined. As in previous examples, the effectivity index is close to unity, indicating that the estimator can be successfully employed for skewed meshes.

Primal dofs	$Q (u, q) - Q (u^{h}, q^{h})$	Dual dofs	${^η}_{e s t}$	$_{e f f}$
867	1.9480e-01	1875	2.5397e-01	1.304
3267	3.9104e-02	7203	4.6311e-02	1.184
12675	4.0197e-03	28227	4.2989e-03	1.069
49923	1.7641e-04	111747	1.7937e-04	1.017
198147	5.0934e-06	444675	5.0988e-06	1.001

Table 5: Results for the convection-dominated diffusion problem (30), with QoI (20) for uniform mesh refinements of a skewed initial mesh partition and

P e = 200

4.2.3 Raviart-Thomas Approximation Of Fluxes

Until this point, we have used $C^{0}$ approximations for both trial variables, as our experience has shown this to yield good approximations Calo et al. (2018). Because $q \in H (div, Ω)$ , the $C^{0}$ approximations have an overly restrictive regularity. Commonly, in mixed FE methods, Raviart-Thomas rather than $C^{0} (Ω)$ approximations are used. To show our approach also provides reliable estimates for such approximations we now consider the case in which we use a Raviart-Thomas approximation Raviart and Thomas (1977); Girault and Raviart (1986); Brezzi and Fortin (1991) for the variables in $H (div, Ω)$ (i.e., $q$ and $r$ ). We again consider the case where $b = {1, 1}^{T}$ , $P e = 100$ , and choose the same QoI (20), i.e., the average solution $u$ in the top right quadrant of the unit square. To approximate the error in the QoI, we now use tetrahedral elements in which $u^{h}, p^{h}$ are discretized with $C^{0}$ polynomials, while $q^{h}, r^{h}$ are discretized using Raviart-Thomas bases. The initial mesh consist of two triangles which are refined uniformly after each computation using orders of approximation of 2 for the primal and 3 for the dual problems, respectively.

For the same element partition, we achieve slightly higher accuracy at a slightly lower number of degrees of freedom for $C^{0}$ approximations versus Raviart-Thomas approximations for $q^{h}$ , as evident from Tables 6 and 7. Comparison of the results in these tables also reveal that there is no significant difference between the two approximations in terms of the accuracy of the error estimates. Raviart-Thomas approximations are used in mixed FE methods as they result in stable FE approximations. Contrarily, $C^{0}$ approximations for $H (div, Ω)$ variables cannot be employed in the same straightforward manner for mixed FE methods and will lead to a violation of discrete inf-sup conditions Brezzi and Fortin (1991). However, in the AVS-FE method, this problem is avoided by employing the DPG philosophy and optimal test functions that ensure the discrete inf-sup condition.

Primal dofs	$Q (u, q) - Q (u^{h}, q^{h})$	$∥ q - q^{h} ∥_{L^{2} (Ω)}$	Dual dofs	${^η}_{e s t}$	$_{e f f}$
243	3.1381e-01	8.3021e-02	507	1.7312e-01	0.552
867	1.9449e-01	5.7260e-02	1875	1.5716e-01	0.808
3267	7.3123e-02	3.3955e-02	7203	7.2499e-02	0.991
12675	1.3955e-02	1.4723e-02	28227	1.4085e-02	1.009
49923	1.4397e-03	4.6769e-03	111747	1.4432e-03	1.002

Table 6: Results for the convection-dominated diffusion problem (30), with QoI (20) for uniform mesh refinements with a

C^{0}

approximation for both variables and

P e = 100

Primal dofs	$Q (u, q) - Q (u^{h}, q^{h})$	$∥ q - q^{h} ∥_{L^{2} (Ω)}$	Dual dofs	${^η}_{e s t}$	$_{e f f}$
257	3.5772e-01	9.0700e-02	529	1.8006e-01	0.503
961	2.3563e-01	6.1567e-02	2017	1.7260e-01	0.733
3713	9.9548e-02	3.7049e-02	7873	9.5166e-02	0.956
14593	2.0993e-02	1.7127e-02	31105	2.1050e-02	1.003
57857	2.2705e-03	5.7263e-03	123649	2.2741e-03	1.001

Table 7: Results for the convection-dominated diffusion problem (30), with QoI (20) for uniform mesh refinements with a Raviart-Thomas approximation of fluxes and

P e = 100

5 Adaptive Mesh Refinement

To demonstrate application of the new alternative error estimate (18) and the resulting error indicators (28) in an $h$ -adaptive process, we use the same form of our model problem, i.e., $b = {1, 1}^{T}$ and $P e = 100$ . As the adaptive strategy for goal oriented mesh refinement we use the method by Oden and Prudhomme Oden and Prudhomme (2001), i.e.,

if \frac{| ϵ_{m} |}{{max}_{K_{m} \in P_{h}} | ϵ_{m} |} > δ, then refine element m,

(33)

where $δ$ is the tolerance for refinement, i.e., $0 < δ < 1$ . In the following numerical verification, we pick $δ = 0.5$ . The QoI we consider is again the average of $u$ in the upper right quadrant (see (20)). The primal problem is approximated using $C^{0}$ continuous polynomials of degree 2, whereas the dual problem is approximated using $p + 1 = 3$ .

The initial mesh consists of 2 triangular elements and is too coarse to resolve the boundary layers in both primal and dual solutions leading to poor error indicators. This effect is shown in Figure 6, where the error indicators are largest in the corner of the dual boundary layer which would result in mesh refinements at the ’wrong’ location. To avoid initial mesh refinements that are poorly suited to reduce the error in the QoI, we initially perform uniform mesh refinements until the error estimate ${^η}_{e s t}$ begins to decrease and indicate that the error indicators $ϵ_{m}$ (28) have become reliable.

Figure 6: Element-wise distribution of the error indicators $ϵ_{m}$ (28) on a coarse mesh.

This can for example be seen in Figure 7, where we show the solution and element-wise error indicators on a uniform mesh of $16 \times 16$ elements, the last mesh that has been uniformly refined and ${^η}_{e s t}$ started to decrease. Here, we see that indicators in corner of the primal boundary layer are now of a magnitude that result in local mesh refinements in the right locations.

In Figure 8, we show the error indicators for an intermediate (step 9) and the final (step 18) step of the adaptive process. In both these cases, the mesh has been refined such that the boundary layers in both primal and dual problems are sufficiently resolved to yield error indicators that are highest in the region of the QoI. The corresponding final adapted mesh is shown in Figure 9. As expected from the current choice of QoI, the mesh refinements have been focused near the primal boundary layer and the QoI.

Figure 9: Final mesh of the goal-oriented $h -$ adaptive refinements.

Lastly, we present the convergence history of the error estimator ${^η}_{e s t}$ and the effectivity index $_{e f f}$ in Figure 10. The plot of estimated error and $∥ u - u^{h} ∥_{L^{2} (Ω)}$ in Figure 10(a) shows that while the adaptive process ensures a small error in the QoI, the global error in the $L^{2} (Ω)$ norm is several orders of magnitude larger as expected since the adaptive process is targeting to reduce the error in the QoI rather than $∥ u - u^{h} ∥_{L^{2} (Ω)}$ . The effectivity index shown in Figure 10(b) demonstrates that the proposed alternative approach to goal-oriented error estimation delivers highly accurate estimates even when the error becomes very small.

(a) ${^η}_{e s t}$ and $∥ u - u^{h} ∥_{L^{2} (Ω)}$ .

6 Conclusions

We have presented goal-oriented a posteriori error estimates for the AVS-FE method. This method is a hybrid Petrov-Galerkin method which uses classical $C^{0} (Ω)$ or Raviart-Thomas FE trial basis functions, while the test space consist of functions that are discontinuous across element edges. The broken topology of the test space allows us to employ the DPG philosophy and compute optimal test functions element-by-element, i.e., completely locally. In an effort to derive a posteriori error estimates of the AVS-FE computations we have introduced two types goal-oriented error estimates. The first estimate follows the classical approach of Becker and Rannacher (2001) where, by duality, the dual solution is sought in the test space $V$ , which in the case of the AVS-FE method is a broken Hilbert space. However, we show that through numerical verifications of the classical Laplace BVP that this approach yields error estimates with poor accuracy. To resolve this, we introduce a second estimate based on consideration of the PDE that governs the dual solution. The estimate is then established by computing $C^{0} (Ω)$ or Raviart-Thomas AVS-FE approximations of this PDE. Numerical verifications involving pure diffusion as well as convection-dominated diffusion problems show that the new alternative error estimate is capable of accurately predicting errors for different QoIs and mesh partitions.

In order to employ the new a posteriori error estimation methodology in mesh adaptivity we also here derived error indicators to guide any $h -$ adaptive process. The error indicators essentially are the element-wise restriction of the residual operator (28). Numerical verifications show that when the error indicators used with classical refinement strategies Oden and Prudhomme (2001), they lead to a mesh adaptive process able to reduce the error in the QoI within a defined accuracy while at the same time delivering highly accurate predictions of the error in the QoI even when the error is small. In a previous paper Calo et al. (2018), the results presented are all computed using $C^{0}$ approximations for both trial variables. Here, we also presented results in which the fluxes are computed by using Raviart-Thomas approximations. These results indicate that $C^{0}$ approximations yield results that are of the same quality in terms of both error estimation, and result in slightly higher accuracy for the flux variable at a slightly lower number of degrees of freedom. Hence, the use of $C^{0}$ approximations for both variables remains attractive due to its lower computational cost, and ease of implementation in existing FE software.

In a forthcoming paper, we intend to extend the AVS-FE method and the new alternative error estimates to other problems such as the nonlinear Cahn-Hilliard equation.

Acknowledgements

The numerical verifications presented in Section 4.2.3 were computed using the computational framework Firedrake Rathgeber et al. (2017) and the $h -$ adaptive process presented in Section 5 was implemented in the computational framework FEniCS Alnæs et al. (2015).

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Goal-Oriented Error Estimation for the Automatic Variationally Stable FE Method for Convection-Dominated Diffusion Problems

A Stable FE Method For the Space-Time Solution of the Cahn-Hilliard Equation

An Unconditionally Stable Space-Time FE Method for the Korteweg-de Vries Equation

Automatic Variationally Stable Analysis for Finite Element Computations: Transient Convection-Diffusion Problems

A Petrov-Galerkin method for nonlocal convection-dominated diffusion problems

An automatic-adaptivity stabilized finite element method via residual minimization for heterogeneous, anisotropic advection-diffusion-reaction problems

A Stable Mixed FE Method for Nearly Incompressible Linear Elastostatics

Generalized Prager-Synge Inequality and Equilibrated Error Estimators for Discontinuous Elements

1 Introduction

2 Variationally Stable Analysis for Finite Element Computations

2.1 Model Problem and Notation

2.2 The AVS-FE Weak Formulation

Lemma 2.0

Remark 2.0

2.3 AVS-FE Discretization

Remark 2.0

Remark 2.0

Remark 2.0

3 Goal-Oriented Error Estimation - Classical Approach

Lemma 3.0

Remark 3.0

4 Goal-Oriented Error Estimation - Alternative Approach

Remark 4.0

4.1 Numerical Verification - Diffusion Problem

4.2 Numerical Verifications - Convection-Dominated Diffusion

4.2.1 Uniform Meshes

4.2.2 Non-Uniform Mesh

4.2.3 Raviart-Thomas Approximation Of Fluxes

5 Adaptive Mesh Refinement

6 Conclusions

Acknowledgements

References

Goal-Oriented Error Estimation for the Automatic Variationally Stable FE Method for Convection-Dominated Diffusion Problems

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1 Introduction

2 Variationally Stable Analysis for Finite Element Computations

2.1 Model Problem and Notation

2.2 The AVS-FE Weak Formulation

Lemma 2.0

Remark 2.0

2.3 AVS-FE Discretization

Remark 2.0

Remark 2.0

Remark 2.0

3 Goal-Oriented Error Estimation - Classical Approach

Lemma 3.0

Remark 3.0

4 Goal-Oriented Error Estimation - Alternative Approach

Remark 4.0

4.1 Numerical Verification - Diffusion Problem

4.2 Numerical Verifications - Convection-Dominated Diffusion

4.2.1 Uniform Meshes

4.2.2 Non-Uniform Mesh

4.2.3 Raviart-Thomas Approximation Of Fluxes

5 Adaptive Mesh Refinement

6 Conclusions

Acknowledgements

References