Analysis of non-conforming DPG methods on polyhedral meshes using fractional Sobolev norms

Highlights

• Variational formulations in fractional spaces of a model problem are analyzed.

• Well-posedness of the broken ultraweak formulation in fractional spaces is proven.

• A DPG finite element discretization of the latter is shown to be stable.

• A-priori error estimates for this DPG method on polyhedral meshes are derived.

• Numerical results with polyhedral meshes and high-order approximation are provided.

Abstract

The work is concerned with two problems: (a) analysis of a discontinuous Petrov–Galerkin (DPG) method set up in fractional energy spaces, (b) use of the results to investigate a non-conforming version of the DPG method for general polyhedral meshes. We use the ultraweak variational formulation for the model Laplace equation. The theoretical estimates are supported with 3D numerical experiments.

Introduction

The presented work is concerned with the analysis of a non-conforming version of the Discontinuous Petrov–Galerkin (DPG) Method with Optimal Test Functions based on the Ultraweak (UW) variational formulation. We shall focus on the model Poisson problem,

$$\left\{\begin{array}{ccc}\hfill -\Delta u& =f\hfill & \text{in }\Omega \hfill \\ \hfill u& =0\hfill & \text{on }\Gamma \hfill \end{array}\right.$$

where $\Omega \subset {\mathbb{R}}^N$ is a Lipschitz domain with boundary $\Gamma$, and $N=2,3$. The UW formulation derives from the equivalent system of first order equations,

$$\left\{\begin{array}{ccc}\hfill \sigma -\nabla u& =0\hfill & \text{in }\Omega \hfill \\ \hfill -\mathrm{div}\sigma & =f\hfill & \text{in }\Omega \hfill \\ \hfill u& =0\hfill & \text{on }\Gamma \hfill \end{array}\right.$$

which can be represented concisely in the operator form

$$A\mathsf{u}=\mathsf{f}.$$

Here $\mathsf{u}≔(\sigma ,u)$ is a group variable, $\mathsf{f}=(0,f)$, and

$$A\mathsf{u}=A(\sigma ,u)=(\sigma -\nabla u,-\mathrm{div}\sigma ).$$

More precisely,

$$A:{\mathsf{L}}^2\left(\Omega \right)\supset D\left(A\right)\to {\mathsf{L}}^2\left(\Omega \right)$$

is a closed operator with

$$D\left(A\right)≔\{(\sigma ,u)\in {\mathsf{L}}^2\left(\Omega \right);A(\sigma ,u)\in {\mathsf{L}}^2\left(\Omega \right),u=0\text{ on }\Gamma \}=H(\mathrm{div},\Omega )\times H_0^1\left(\Omega \right),$$

where ${\mathsf{L}}^2\left(\Omega \right)≔L^2{\left(\Omega \right)}^N\times L^2\left(\Omega \right)$. The UW variational formulation is obtained by multiplying both equations with test functions $\tau ,v$ and integrating by parts both equations. We obtain

$$\left\{\begin{array}{c}\mathsf{u}\in {\mathsf{L}}^2\left(\Omega \right)\hfill \\ (\mathsf{u},A^{\ast }\mathsf{v})=(\mathsf{f},\mathsf{v})\hfill & \mathsf{v}\in D\left(A^{\ast }\right).\hfill \end{array}\right.$$

Here $A^{\ast }$ is the ${\mathsf{L}}^2$-adjoint of operator $A$,

$$\begin{array}{c}{A}^{\ast }:{\mathsf{L}}^2\left(\Omega \right)\supset D\left(A^{\ast }\right)\to {\mathsf{L}}^2\left(\Omega \right),A^{\ast }\mathsf{v}=A(\tau ,v)=(\tau +\nabla v,\mathrm{div}\tau )\in {\mathsf{L}}^2\left(\Omega \right),\hfill \\ D\left(A^{\ast }\right)=D\left(A\right).\hfill \end{array}$$

The broken UW formulation is obtained by testing with test functions from a larger space of broken test functions $V\left({\mathcal{T}}_h\right)≔H(\mathrm{div},{\mathcal{T}}_h)\times H^1\left({\mathcal{T}}_h\right)$ where

$$H(\mathrm{div},{\mathcal{T}}_h)≔\{\tau \in L^2{\left(\Omega \right)}^N:\tau |K\in H(\mathrm{div},K)\forall K\in {\mathcal{T}}_h\},$$$$H^1\left({\mathcal{T}}_h\right)≔\{v\in L^2\left(\Omega \right):v|K\in H^1\left(K\right)\forall K\in {\mathcal{T}}_h\}.$$

The symbol ${\mathcal{T}}_h$ denotes an open partition of $\Omega$ into Lipschitz finite elements of maximum size $h>0$, whose set of interfaces (or skeleton) is ${\Gamma }_h$. This formulation necessitates introducing additional unknowns — the Lagrange multipliers $({\hat{\sigma }}_n,\hat{u})$, called traces,

$$\left\{\begin{array}{c}\mathsf{u}\in {\mathsf{L}}^2\left(\Omega \right),\hat{\mathsf{u}}≔({\hat{\sigma }}_n,\hat{u})\in H^{-1∕2}\left({\Gamma }_h\right)\times {\tilde{H}}^{1∕2}\left({\Gamma }_h\right)\hfill \\ (\mathsf{u},A_h^{\ast }\mathsf{v})-{〈\hat{\mathsf{u}},\mathsf{v}〉}_{{\Gamma }_h}=(\mathsf{f},\mathsf{v})\mathsf{v}\in V\left({\mathcal{T}}_h\right).\hfill \end{array}\right.$$

Here the operator $A_h^{\ast }:V\left({\mathcal{T}}_h\right)\to {\mathsf{L}}^2\left(\Omega \right)$ is the extension of the adjoint operator $A^{\ast }$, defined elementwise by

$$\left(A_h^{\ast }(\tau ,v)\right){|}_K≔(\tau {|}_K+\nabla \left(v{|}_K\right),\mathrm{div}\left(\tau {|}_K\right)).$$

The duality pairing on the skeleton present in (1.2) is given by

$${〈\hat{\mathsf{u}},\mathsf{v}〉}_{{\Gamma }_h}=\sum _{K\in {\mathcal{T}}_h}\left[{〈{\hat{\sigma }}_n,v〉}_{\partial K}+{〈\hat{u},\tau \cdot n〉}_{\partial K}\right].$$

Trace ${\hat{\sigma }}_n$ is defined by taking a function from $H(\mathrm{div},\Omega )$, restricting it to each element $K\in {\mathcal{T}}_h$, and taking its normal trace on element boundary $\partial K$. Similarly, trace $\hat{u}$ is obtained by taking a function from $H_0^1\left(\Omega \right)$, restricting it to every $K\in {\mathcal{T}}_h$, and taking its trace on $\partial K$. The “tilde” over ${\tilde{H}}^{1∕2}\left({\Gamma }_h\right)$ hides the boundary condition on $\hat{u}$, that is,

$${\tilde{H}}^{1∕2}\left({\Gamma }_h\right)≔\{\hat{u}\in H^{1∕2}\left({\Gamma }_h\right):\hat{u}=0\text{ on }\Gamma \}.$$

It has been shown in [1] that problem (1.2) is well-posed.

The broken UW formulation is perhaps the least demanding formulation in terms of global conformity for a finite element (FE) method. The $L^2$ space is discretized with discontinuous functions, and so are the broken test spaces. The trace space $H^{-1∕2}\left({\Gamma }_h\right)$ may be discretized with discontinuous functions over faces in the mesh skeleton ${\Gamma }_h$. Only the discretization of $\hat{u}$, done with traces of $H^1$-conforming elements, requires global continuity over ${\Gamma }_h$.

If ${\mathcal{T}}_h$ consists of general polyhedral elements, instead of the standard element shapes, the conforming discretization of $\hat{u}$ becomes really challenging, because we need to have basis functions that are continuous over a skeleton of general polygonal faces. One alternative is to drop the continuity requirement, and discretize that variable with discontinuous polynomials, thus getting an easier way to discretize at the expense of losing $H^{1∕2}$-conformity. The good news is that, just by perturbing our Sobolev spaces by $ϵ>0$, these discontinuous trace functions become conforming. More precisely, if $w:{\mathbb{R}}^{N-1}\to \mathbb{R}$ is bounded, of compact support, and piecewise smooth – possibly discontinuous – then $w\in H^{1∕2-ϵ}\left({\mathbb{R}}^{N-1}\right)$ (for illustrative examples see [2, Exercise 3.22] and [3, Exercise 1.39]).

The goal of this paper is to carry out the stability analysis and derive a-priori error estimates for a non-conforming version of both the ideal and practical DPG methods for the Poisson problem set up in fractional Sobolev spaces, in which the discretization of trace $\hat{u}$ is done with discontinuous polynomials.

A series of numerical experiments involving various polyhedral meshes constitute a substantial part of the work. Given the complexity of a computational implementation of fractional norms, all numerical results reported below correspond to the original functional setting, which can be interpreted as the limit case of the fractional setting considered in our theoretical framework.

In order to avoid an excessive length in the document, some information related to this work has been left out. We believe nonetheless that the present version remains self-contained. As a supplement to this paper, a more extensive set of numerical results, along with an Appendix dealing with Fortin operators and a more thorough literature review can be found in the technical report [4].

The numerical approximation to the solution of (1.2) requires a non-symmetric functional setting and answering the question of whether the discrete version of the problem is well posed and stable. To this end, the DPG method, originally introduced in [5], offers a framework to solve this model problem. The starting point is an infinite-dimensional variational problem that satisfies the inf–sup condition. For a given discrete trial space, the DPG machinery provides the optimal test subspace that preserves that inf–sup condition, making the discrete problem automatically stable. All of these details are not presented here for the sake of the document length, but we encourage those readers still unfamiliar with DPG to refer to [6], Chapter 5. All the necessary concepts that we apply below are well explained therein, especially the differences between the ideal and practical versions of DPG, the implications of the so-called Fortin operator, and the role of the test norm and Gram matrix in the computational implementation.

The solution of boundary value problems through numerical methods that support polygonal or polyhedral meshes has attracted rising attention over the last two decades. By generalizing those methods to non-standard element shapes, it may be possible to enable rapid engineering analysis through the use of novel meshing approaches, while circumventing some common issues of standard meshing [7]. Such interest is also often motivated by specific applications, for which the use of such meshes is more recommended than that of standard meshes, or has delivered better results. To have a glance on a few of those applications and the existing methods that support polytopal meshes, please refer to the brief overview on the subject presented in the introduction of [4], [8].

In view of the many possibilities that can be enabled through polytopal meshes, DPG has been recently introduced into the family of polygonal methods by Vaziri, Fuentes, Mora and Demkowicz [8]. In 2D, this methodology, labeled as PolyDPG, has been enabled by the broken UW formulation. In particular, the trace $\hat{u}$ can be discretized with continuous functions over the skeleton of a 2D mesh. The highlights of PolyDPG in 2D, in contrast to other methods, are that it is a high-order and stable polygonal FE method, whose ultraweak-conforming discrete spaces are constructed with polynomials only.

On general polyhedral meshes in 3D, as mentioned above, opting for a discontinuous basis is a practical but non-conforming way to discretize the space for unknown $\hat{u}$. For this reason, we must study the effect on the stability and approximability of the PolyDPG method when choosing this discontinuous alternative. In the literature there are results concerning this situation in DPG with standard elements, but when the element is an arbitrary polyhedron the analysis turns more complicated, starting with the fact that there is no unique reference element or face, so that we need to argue that the error estimates can be independent of the element shape.

The first computational results concerning a non-conforming DPG method are found in an early technical report by Demkowicz and Gopalakrishnan [9]. An analysis on why this implementation works has been developed by Heuer, Karkulik and Sayas in [10]. The abstract procedure therein outlined is followed by us to obtain a-priori error estimates (see Section 5). Another relevant development in non-conforming DPG is that by Ernesti and Wieners on space–time DPG for linear wave problems [11]. For a more complete review on these previous works please see [4].

We now explain the organization of the article. In Section 2, we start by analyzing the infinite-dimensional scenario of the Poisson model problem formulated in fractional spaces. Firstly, we show that the classical variational formulation is well-posed, which in turn helps proving that the UW variational formulation is well-posed too, as derived in Section 3. In Section 4, the idea of localization of norms leads to both the inf–sup condition of the broken UW formulation, and the stability for the ideal DPG with fractional norms. Section 5 deals with the implications of the present analysis on the practical non-conforming DPG. This is followed by the study on approximability of the discrete trace spaces (Section 6). The ultimate a-priori error estimates herein developed are found in Section 7. A collection of numerical results of PolyDPG with discontinuous traces is presented in Section 8. The conclusions are summarized in Section 9.