Analysis of non-conforming DPG methods on polyhedral meshes using fractional Sobolev norms
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, where is a Lipschitz domain with boundary , and . The UW formulation derives from the equivalent system of first order equations, which can be represented concisely in the operator form Here is a group variable, , and More precisely, is a closed operator with where . The UW variational formulation is obtained by multiplying both equations with test functions and integrating by parts both equations. We obtain Here is the -adjoint of operator , The broken UW formulation is obtained by testing with test functions from a larger space of broken test functions where The symbol denotes an open partition of into Lipschitz finite elements of maximum size , whose set of interfaces (or skeleton) is . This formulation necessitates introducing additional unknowns — the Lagrange multipliers , called traces, Here the operator is the extension of the adjoint operator , defined elementwise by The duality pairing on the skeleton present in (1.2) is given by Trace is defined by taking a function from , restricting it to each element , and taking its normal trace on element boundary . Similarly, trace is obtained by taking a function from , restricting it to every , and taking its trace on . The “tilde” over hides the boundary condition on , that is, 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 space is discretized with discontinuous functions, and so are the broken test spaces. The trace space may be discretized with discontinuous functions over faces in the mesh skeleton . Only the discretization of , done with traces of -conforming elements, requires global continuity over .
If consists of general polyhedral elements, instead of the standard element shapes, the conforming discretization of 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 -conformity. The good news is that, just by perturbing our Sobolev spaces by , these discontinuous trace functions become conforming. More precisely, if is bounded, of compact support, and piecewise smooth – possibly discontinuous – then (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 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 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 . 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.
Section snippets
Equivalence of different fractional norms
We will move rather freely between different definitions of fractional Sobolev spaces and the corresponding norms. Let be a Lipschitz domain. For any , we define the space as the space of restrictions from with the minimum energy extension norm. Spaces are defined using Bessel potentials, see [2, p. 77], or [12, Section 3.1].
For , we define the space where is the Sobolev space of integral order and
Ultraweak variational formulation
The well-posedness analysis of various variational formulations presented in [20] extends easily to fractional spaces. Let . Operator corresponding to the strong formulation is defined as follows: Its topological transpose is defined on the dual space: We assume that , which guarantees the
Broken UW variational formulation
From here on, we need to introduce quasi-uniformity and shape-regularity assumptions for the mesh. It is well known how the two conditions are defined for standard FE. For more general polyhedral elements, the quasi-uniformity assumption on the mesh remains unaltered. However, the shape regularity needs to be understood differently. For the sake of Lemma 2, what matters is to have a uniform bound on the number of immediate neighbors an element can have. An effective way to bound such a quantity
Implications for the non-conforming DPG method
In the preceding sections we have developed a theory for a DPG method formulated in fractional spaces. By weakening the trial norm and strengthening the test norm, we have been able to employ the discontinuous discretization of traces which, in the relaxed energy setting, has become conforming. The discussed results require computations with the stronger, fractional test norm which is not attractive at all from the practical point of view.
In this section, we show how one can combine arguments
Best approximation error estimates for polyhedral elements
In this section we develop best approximation error estimates corresponding to the discrete spaces with which we are discretizing the trial variables. We work with piecewise polynomial spaces defined at every element in or face in , without any requirement of global continuity. Let us fix an integer . Thus, for field variables , at every element we use polynomials of degree , while for the traces we define the polynomials over each face, having degree for and degree
Polyhedra with triangular or quadrilateral faces.
We recall first the standard way to approximate the best approximation error for traces in the minimum energy extension norms under the assumption that trace is discretized with trace of standard, -conforming finite element function . For the -traces we have: with independent of element and function .
A similar argument holds for the normal trace:
About the implementation
This section exclusively considers the three-dimensional case (). A modified version of the -adaptive finite element code by Demkowicz et al. (see [24]) has been implemented in order to support high-order approximation with (simple) polyhedral elements of arbitrary number of vertices and (flat) faces. The new code is able to obtain the practical DPG solution of any well-posed broken ultraweak variational formulation, but with discontinuous discrete traces. However, when all faces in
Conclusions
We have presented a stability and convergence analysis for a class of DPG methods based on the ultraweak (UW) variational formulation for general polyhedral meshes (PolyDPG) for a model Poisson problem in 3D. The DPG methodology based on the UW formulation offers the most relaxed setting in terms of conformity. Only one variable - the trace, defined on the mesh skeleton, needs to be continuous in order to guarantee a fully conforming discretization based on the standard energy spaces
CRediT authorship contribution statement
Constantin Bacuta: Conceptualization, Methodology, Formal analysis, Writing - review & editing. Leszek Demkowicz: Conceptualization, Methodology, Formal analysis, Writing - original draft, Writing - review & editing. Jaime Mora: Conceptualization, Software, Visualization, Writing - original draft, Writing - review & editing. Christos Xenophontos: Conceptualization, Methodology, Writing - review & editing.
Acknowledgments
L. Demkowicz and J. Mora were partially supported with National Science Foundation grant No. 1819101 and the Sandia National Laboratory exploratory LDRD project No. 19-1038, whose PI was Dr. Mohamed Ebeida, to whom we are deeply thankful. The first and last authors are grateful for an Oden Institute fellowship which allowed them to visit UT Austin in the Fall of 2019. C. Bacuta’s work was also supported by National Science Fourndation grant No. 2011615.
The authors appreciate the detailed
References (33)
- et al.
Breaking spaces and forms for the DPG method and applications including Maxwell equations
Comput. Math. Appl.
(2016) - et al.
A class of discontinuous Petrov–Galerkin methods. Part I: The transport equation
Comput. Methods Appl. Mech. Engrg.
(2010) - et al.
High-order polygonal discontinuous Petrov–Galerkin (PolyDPG) using ultraweak formulations
Comput. Methods Appl. Mech. Engrg.
(2018) - et al.
Note on discontinuous trace approximation in the practical DPG method
Comput. Math. Appl.
(2014) - et al.
Hitchhiker’s guide to the fractional sobolev spaces
Bull. Sci. Math.
(2012) - et al.
And -conforming projection-based interpolation in three dimensions. Quasi-optimal -interpolation estimates
Comput. Methods Appl. Mech. Engrg.
(2005) - et al.
De Rham diagram for finite element spaces
Comput. Math. Appl.
(2000) - et al.
Orientation embedded high order shape functions for the exact sequence elements of all shapes
Comput. Math. Appl.
(2015) - et al.
On the flexibility of agglomeration based physical space discontinuous Galerkin discretizations
J. Comput. Phys.
(2012) Strongly Elliptic Systems and Boundary Integral Equations
(2000)
p-and hp-Finite Element Methods: Theory and Applications in Solid and Fluid Mechanics
Analysis of Non-Conforming DPG Methods on Polyhedral Meshes using Fractional Sobolev NormsTechnical Report 03
Lecture Notes on Mathematical Theory of Finite ElementsTechnical Report 11
Applications of polyhedral finite elements in solid mechanics
Analysis of the DPG method for the Poisson problem
SIAM J. Numer. Anal.
A space-time discontinuous Petrov–Galerkin method for acoustic waves
Cited by (5)
Combining DPG in space with DPG time-marching scheme for the transient advection–reaction equation
2022, Computer Methods in Applied Mechanics and EngineeringCitation Excerpt :The authors proved optimal convergence and approximation properties comparing to the classical Discontinuous-Galerkin (DG) method. Since then, the DPG method has been applied to a wide variety of problems [2–7] and has been analyzed over the years by several authors [8–11]. Recently, Demkowicz and Roberts revisited the method for advection–reaction problems in [12].
Recent Advances in Least-Squares and Discontinuous Petrov–Galerkin Finite Element Methods
2021, Computers and Mathematics with ApplicationsA DISCONTINUOUS PETROV–GALERKIN METHOD FOR REISSNER–MINDLIN PLATES
2023, SIAM Journal on Numerical AnalysisA DPG method for Reissner-Mindlin plates
2022, arXiv