A quasi-optimal test norm for a DPG discretization of the convection-diffusion equation

Abstract

In this work, we propose a new quasi-optimal test norm for a discontinuous Petrov-Galerkin (DPG) discretization of the ultra-weak formulation of the convection-diffusion equation. We prove theoretically that the proposed test norm leads to bounds between the target norm and the energy norm induced by the test norm which have favorable scalings with respect to the diffusion parameter when compared with existing results for other test norms from the literature. We conclude with numerical experiments to confirm our theoretical results.

Introduction

Let $\mathrm{\Omega }\subset {\mathbb{R}}^d$ be a bounded polyhedral domain and consider the problem

$$\mathrm{\nabla }\cdot (\mathbf{a}u-\epsilon \mathrm{\nabla }u)=f\text{in }\mathrm{\Omega },u=0\text{on }{\mathrm{\Gamma }}_D,(\mathbf{a}u-\epsilon \mathrm{\nabla }u)\cdot \mathit{n}=0\text{on }{\mathrm{\Gamma }}_N,$$

with n denoting the outward unit normal to the boundary ∂Ω and where ${\mathrm{\Gamma }}_D$ and ${\mathrm{\Gamma }}_N$ are two disjoint open subsets satisfying ${\bar{\mathrm{\Gamma }}}_D\cup {\bar{\mathrm{\Gamma }}}_N=\partial \mathrm{\Omega }$ and ${\mathrm{\Gamma }}_N\subseteq {\mathrm{\Gamma }}_{in}:=\{x\in \partial \mathrm{\Omega }|\mathbf{a}\cdot \mathbf{n}\le 0\}$. Here, the data satisfies $\epsilon \in {\mathbb{R}}^{+}$, $\mathbf{a}\in {[C(\overline{\mathrm{\Omega }})]}^d$ with $C(\overline{\mathrm{\Omega }})\ni \mathrm{\nabla }\cdot \mathbf{a}\ge 0$ and $f\in L^2(\mathrm{\Omega })$.

The solution to problem (1.1) is difficult to approximate numerically due to the presence of boundary/interior layers – areas of steep solution gradients with a width dependent upon ε. It is well known that regardless of what formulation of problem (1.1) is used that discretizing it using the standard finite element method results in spurious oscillations occurring in the numerical solution near the layers until a sufficient number of elements have been added locally with the problem worsening as $\epsilon \to 0^{+}$; for this reason, multiple stable schemes have been developed for problem (1.1) over the years. The discontinuous Galerkin (DG) method originally developed by Reed/Hill for the neutron transport equation [28] has been successfully applied to the convection-diffusion equation through a variety of different stable discretizations [1], [3], [13], [29]. The so-called hybridizable discontinuous Galerkin (HDG) methods are another popular class of schemes used to discretize the convection-diffusion equation, cf., [12], [19], [23], [26], [27]. We also cannot discuss numerical methods for convection-diffusion equations without mentioning the highly successful streamline-upwind Petrov-Galerkin (SUPG) method [5], [6]; the SUPG method is unique among all of these previously mentioned methods in the sense that it uses standard conforming finite elements for the trial space but uses a special space of test functions, biased in the upwind direction, to impart the method its stability [10]. This idea of test functions imparting stability to numerical methods in fact dates back to [21] and it is this critical observation that lies at the heart of the discontinuous Petrov-Galerkin method. When discussing Petrov-Galerkin methods, we should also mention [4] which is unusual among numerical methods in that it allows passing to the convective limit.

The discontinuous Petrov-Galerkin (DPG) method was originally introduced by Demkowicz/Gopalakrishnan in the context of the transport equation [16]. This idea was then abstracted and applied to a variety of different equations [14], [15], [32]. A thorough analysis of the DPG method, as applied to the Poisson equation, also exists [17]. The DPG method can be applied to the abstract variational formulation of finding $u\in U$ such that

$$B(u,v)=l(v)\forall v\in V,$$

where U and V are some Hilbert Spaces, $B(\cdot ,\cdot ):U\times V\to \mathbb{R}$ is a continuous bilinear form and $l:V\to \mathbb{R}$ is a continuous linear functional. As is common, for the discretization of (1.2), we seek $u_h\in U_h\subset U$ such that

$$B(u_h,v_h)=l(v_h)\forall v_h\in V_h\subset V,$$

$\dim (U_h)=\dim (V_h)<\infty$. In general, the existence of a solution to the continuous problem (1.2) does not guarantee that the discrete problem (1.3) also has a solution unless $V_h$ is chosen to be the so-called space of optimal test functions. To be specific, (1.3) becomes the (theoretical) DPG method when the test space is chosen to be the space of optimal test functions $V_h:=T(U_h)$ where $T:U\to V,u\mapsto Tu$ is the trial to test operator given by the unique solution of the variational problem

$${(Tu,v)}_V=B(u,v)\forall v\in V.$$

Here, the inner product ${(\cdot ,\cdot )}_V$ induces a norm $||\cdot |{|}_V$ referred to as the test norm which, effectively, defines the DPG method via the variational equation above. Note that the use of this test space, in addition to guaranteeing the solvability of (1.3), also means that $u_h\in U_h$ solving (1.3) gives the best approximation error in the energy norm $||\cdot |{|}_E$, i.e.,

$$||u-u_h|{|}_E:=\underset{v\in V}{\sup }\frac{B(u-u_h,v)}{||v|{|}_V}=\underset{w_h\in U_h}{\inf }||u-w_h|{|}_E.$$

We now remark that the space of optimal test functions, as defined above, cannot actually be computed since (1.4) is an infinite dimensional problem; therefore, for the practical DPG method we must approximate this variational equation. Typically, this is done through an enriched test space ${\tilde{V}}_h$ based on the same mesh as $U_h$ but with higher polynomial degree $p+\mathrm{\Delta }p$ so that $\dim ({\tilde{V}}_h)>\dim (U_h)$ though other methods of forming this enriched test space do exist [24], [30]. Thus, if we let $\{{\phi }_i\}$ denote a basis for $U_h$ then we are seeking a basis $\{{\psi }_i\in {\tilde{V}}_h\}$ defining $V_h$ such that

$${({\psi }_i,{\tilde{v}}_h)}_V=B({\phi }_i,{\tilde{v}}_h)\forall {\tilde{v}}_h\in {\tilde{V}}_h.$$

However, this still presents a problem; namely, (1.6) results in a matrix-vector system of higher dimension than that of the original discretization! This difficulty can be overcome by using a discontinuous enriched test space ${\tilde{V}}_h$ and, hence, a discontinuous test space $V_h$. Under the additional assumption that the test norm $||\cdot |{|}_V$ is localizable then (1.6) can now be solved elementwise instead of globally yielding a practical method. Note that $V_h$ must be a subspace of V which means that V must also consist of discontinuous functions; it is necessary to take this into consideration when formulating the bilinear form.

As noted above, the DPG method converges optimally in the energy norm $||\cdot |{|}_E$, however, we usually prefer to measure the error in a norm of our choice $||\cdot |{|}_U$ appropriate to the problem at hand. By duality, $||\cdot |{|}_U$ defines a corresponding norm $||\cdot |{|}_{V,\text{opt}}$ on V referred to as the optimal test norm, viz.,

$$||v|{|}_{V,\text{opt}}:=\underset{u\in U}{\sup }\frac{B(u,v)}{||u|{|}_U}.$$

The discrepancy between the use of $||\cdot |{|}_V$ in the DPG method and the ‘optimal norm’ $||\cdot |{|}_{V,\text{opt}}$ results in a worse convergence bound formulated precisely in the following convergence result from Theorem 2.1 in [32].

Theorem 1.8

Let u and $u_h$ denote the solutions of (1.2) and (1.3), respectively and suppose that the practical norm on V is equivalent to the optimal test norm on V, i.e., that there exists positive constants $C_L$ and $C_U$ such that

$$C_L||v|{|}_V\le ||v|{|}_{V,\mathit{\text{opt}}}\le C_U||v|{|}_V\forall v\in V.$$

Further suppose that the bilinear form B satisfies

$$B(u,v)=0\forall v\in V\Rightarrow u=0.$$

Then

$$||u-u_h|{|}_U\le \frac{C_U}{C_L}\underset{w_h\in U_h}{\inf }||u-w_h|{|}_U.$$

Theorem 1.8 implies that we want to choose the test norm $||\cdot |{|}_V$ to be as close as we can practically get to the optimal test norm $||\cdot |{|}_{V,\text{opt}}$ in order to force $C_L$ and $C_U$ to be as near to unity as possible. Note that we cannot use $||\cdot |{|}_{V,\text{opt}}$ itself in the DPG method since, as we will see below, for practical choices of the norm $||\cdot |{|}_U$ the optimal test norm contains ‘jump norms’ which couple nearby elements meaning it is non-localizable. In the context of convection-diffusion, then, the goal is to find an appropriate trial norm $||\cdot |{|}_U$ in which to measure the error along with a test norm $||\cdot |{|}_V$ which is as close as possible to the optimal test norm thus resulting in a good ratio of the constants $C_U$ to $C_L$. With respect to Theorem 1.8, we also note [9] which shows that the robustness of the field variables is essentially dependent only upon the continuous formulation of the problem and the choice of test norm $||\cdot |{|}_V$.

Most of the DPG discretizations for the convection-diffusion equation are based on the ultra-weak formulation (to be derived in the next section) for which we have a vector-valued test function $\mathit{v}=(v,\mathit{\tau })$ with v corresponding to u and τ corresponding to ∇u. Various different test norms have been proposed for the DPG method as applied to the convection-diffusion equation (1.1). In [18], multiple test norms are trialed including the quasi-optimal test norm

$$||\mathit{v}|{|}_{V,\text{QO}}:=\sqrt{||\mathrm{\nabla }\cdot \mathit{\tau }-\mathbf{a}\cdot \mathrm{\nabla }v|{|}^2+||{\epsilon }^{-1}\mathit{\tau }+\mathrm{\nabla }v|{|}^2+||v|{|}^2},$$

which has performed well in the past on many PDE problems of very different character. Results show that this norm performs well up to a point using the standard p-enriched test space approach but that for higher polynomial degrees it was necessary to use computationally expensive Shishkin-type meshes for the enriched test space [18]. This problem is somewhat remedied in [10], [11] through the use of the mesh-dependent test norm

$$||\mathit{v}|{|}_{V,\text{MD}}:=\sqrt{||C_{v}v|{|}^2+\epsilon ||\mathrm{\nabla }v|{|}^2+||\mathbf{a}\cdot \mathrm{\nabla }v|{|}^2+||C_{\mathit{\tau }}\mathit{\tau }|{|}^2+||\mathrm{\nabla }\cdot \mathit{\tau }|{|}^2},$$

where $C_v{|}_K:=\min \{\sqrt{\epsilon /|K|},1\}$, $C_{\mathit{\tau }}{|}_K:=\min \{1/\sqrt{\epsilon },1/\sqrt{|K|}\}$, $K\in \mathcal{T}$ where $\mathcal{T}$ is some mesh of Ω. It was shown that optimal test functions computed under this norm do not have boundary layers allowing for the standard p-enriched test space approach to work for this norm. All of the works [10], [11], [18] contain upper and lower bounds between $||\mathit{u}|{|}_U$ and $||\mathit{u}|{|}_E$ with which we will compare bounds for our proposed test norm later in this paper. Another work [25] shows that the quasi-optimal test norm

$$||\mathit{v}|{|}_{V,{\text{QO}}_2}:=\sqrt{||\mathrm{\nabla }\cdot \mathit{\tau }-\mathbf{a}\cdot \mathrm{\nabla }v|{|}^2+||{\epsilon }^{-1}\mathit{\tau }+\mathrm{\nabla }v|{|}^2+{\alpha }_1||v|{|}^2+{\alpha }_2||\mathit{\tau }|{|}^2},$$

performs well with the authors choosing the values ${\alpha }_1=1$, ${\alpha }_2={\epsilon }^{-3/2}$ for these parameters in their numerical experiments. As in [18], Shishkin-type meshes are used for the enriched test space in that work.

Here, we propose the mesh-dependent quasi-optimal test norm

$$||\mathit{v}|{|}_V:=\sqrt{||\mathrm{\nabla }\cdot \mathit{\tau }-\mathbf{a}\cdot \mathrm{\nabla }v|{|}^2+{\epsilon }^{-1}||C_{\mathit{\tau }}(\mathit{\tau }+\epsilon \mathrm{\nabla }v)|{|}^2+||v|{|}^2+||\mathrm{\nabla }v|{|}^2},$$

for use in the DPG method approximating (1.1). We prove upper and lower bounds between $||\mathit{u}|{|}_U$ and $||\mathit{u}|{|}_E$ for $||\cdot |{|}_V$ that have favorable scalings with respect to the diffusion parameter ε with minimal assumptions on the convection. Similar upper/lower bounds can also be proven for our proposed test norm $||\cdot |{|}_V$ but with the term $||\mathrm{\nabla }v|{|}^2$ replaced by either ${\epsilon }^{-2}||\mathit{\tau }|{|}^2$ or $||\mathrm{\nabla }\cdot \mathit{\tau }|{|}^2$ just with differing (ε-independent) constants. In light of this observation, we note the similarities between our proposed test norm $||\cdot |{|}_V$ and the quasi-optimal test norm $||\cdot |{|}_{V,{\text{QO}}_2}$ of [25]; due to this, we restrict ourselves to a comparison between the test norms $||\cdot |{|}_V$ and $||\cdot |{|}_{V,\text{MD}}$ only in the numerical experiments section.

The remainder of this paper is organized as follows: in the next section, we derive the ultra-weak variational formulation for (1.1) and in section 3 we introduce the function spaces necessary in order to properly define the ultra-weak variational formulation. In section 4, we prove upper and lower bounds between $||\mathit{u}|{|}_U$ and $||\mathit{u}|{|}_E$ for the proposed test norm (1.12) and compare our results with similar results already in the literature. We then apply our test norm in a variety of numerical experiments in section 5 with a special emphasis on a comparison with the test norm $||\cdot |{|}_{V,\text{MD}}$ from [10], [11] before drawing conclusions in section 6.