CAMPINAS
Instituto de Matemática, Estatística e
Computação Científica
CIRO JAVIER DIAZ PENEDO
High order finite elements with conservative
properties
Elementos finitos de alta ordem com
propriedades de conservação
Campinas
2019
High order finite elements with conservative properties
Elementos finitos de alta ordem com propriedades de
conservação
Tese apresentada ao Instituto de Matemática, Estatística e Computação Científica da Uni-versidade Estadual de Campinas como parte dos requisitos exigidos para a obtenção do título de Doutor em Matemática Aplicada. Thesis presented to the Institute of Mathe-matics, Statistics and Scientific Computing of the University of Campinas in partial ful-fillment of the requirements for the degree of Doctor in Applied Mathematics.
Supervisor: Eduardo Cardoso de Abreu
Este exemplar corresponde à versão
fi-nal da Tese defendida pelo aluno Ciro
Javier Diaz Penedo e orientada pelo
Prof. Dr. Eduardo Cardoso de Abreu.
Campinas
2019
Biblioteca do Instituto de Matemática, Estatística e Computação Científica Ana Regina Machado - CRB 8/5467
Diaz Penedo, Ciro Javier,
D543h DiaHigh order finite elements with conservative properties / Ciro Javier Diaz Penedo. – Campinas, SP : [s.n.], 2019.
DiaOrientador: Eduardo Cardoso de Abreu.
DiaTese (doutorado) – Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica.
Dia1. Método dos elementos finitos. 2. Darcy, Fluxo de. 3. Meios porosos. 4. Equações diferenciais elípticas – Soluções numéricas. 5. Problemas de valores de contorno. I. Abreu, Eduardo Cardoso de, 1974-. II. Universidade Estadual de Campinas. Instituto de Matemática, Estatística e Computação Científica. III. Título.
Informações para Biblioteca Digital
Título em outro idioma: Elementos finitos de alta ordem com propriedades de conservação Palavras-chave em inglês:
Finite element method Darcy flow
Porous media
Elliptic differential equations - Numerical solutions Boundary value problems
Área de concentração: Matemática Aplicada Titulação: Doutor em Matemática Aplicada Banca examinadora:
Eduardo Cardoso de Abreu [Orientador] Marco Lúcio Bittencourt
Juan Carlos Galvis-Arrieta José Augusto Mendes Ferreira Alexandre Loureiro Madureira
Data de defesa: 13-09-2019
Programa de Pós-Graduação: Matemática Aplicada Identificação e informações acadêmicas do(a) aluno(a)
- ORCID do autor: https://orcid.org/0000-0002-8572-6243 - Currículo Lattes do autor: http://lattes.cnpq.br/2471598006296292
pela banca examinadora composta pelos Profs. Drs.
Prof(a). Dr(a). EDUARDO CARDOSO DE ABREU
Prof(a). Dr(a). MARCO LÚCIO BITTENCOURT
Prof(a). Dr(a). JUAN CARLOS GALVIS-ARRIETA
Prof(a). Dr(a). JOSÉ AUGUSTO MENDES FERREIRA
Prof(a). Dr(a). ALEXANDRE LOUREIRO MADUREIRA
A Ata da Defesa, assinada pelos membros da Comissão Examinadora, consta no SIGA/Sistema de Fluxo de Dissertação/Tese e na Secretaria de Pós-Graduação do Instituto de Matemática, Estatística e Computação Científica.
“This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001.”
“This study was also financed in part by the Conselho Nacional de Desenvolvi-mento Científico e Tecnológico - Brasil (CNPq) - Process: 140978/2018-6.”
A pesquisa realizada nesta tese de doutorado é dedicada a introduzir uma nova metodologia de aproximação numérica para problemas de valor de contorno elípticos de segunda ordem, cuja originalidade reside em ótimas propriedades de conservação local de massa. Uma nova análise numérica rigorosa é apresentada para colocar o procedimento em sólida base matemática. Esta nova metodologia é baseada no método de Ritz-Galerkin e permite aproximações localmente conservativas da solução de problemas de valor de contorno elípticos de segunda ordem associados, por exemplo, à dinâmica de fluidos em meios porosos usando espaços de funções polinomiais por partes de alta ordem ou espaços multiescala. Pede-se simplesmente para a solução satisfazer uma determinada restrição de conservação de massa (local) em certos subdomínios. Alternativamente, essa exigência pode ser entendida como uma generalização do Método de Volumes Finitos para espaços funcionais gerais que permitem aproximações conservativas de alta ordem. Com base nessa nova metodologia, introduzimos um novo Método de Elementos Finitos não conformes de alta ordem conservativo (HOCFEM) para problemas de valor de contorno elípticos de segunda ordem. Paralelamente, apresentamos alguns resultados parciais para uma modificação do Método de Elementos Finitos Multiescala Generalizado (GMsFEM) que tem uma íntima conexão com a metodologia abordada no contexto de fluxo em meios porosos. Especificamente, mostramos que a modificação introduzida melhora a eficiência do método em alguns casos. Juntos, HOCFEM and GMsFEM, podem permitir a construção de um solver multiescala eficiente e conservativo para problemas de valor de contorno elípticos de segunda ordem. A nova metodologia é naturalmente paralelizável, embora esta facilidade não seja explorada nesta tese. Além disso, resultados numéricos preliminares foram obtidos combinando HOCFEM com um novo método Lagrangeano-Euleriano para problemas hiperbólicos, a fim de resolver e estudar numericamente o problema fundamental não linear de escoamento bifásico incompressível e imiscível água-óleo de saturação-pressão-velocidade. Experimentos numéricos representativos são apresentados e discutidos com o objetivo de demonstrar a viabilidade das novas metodologias propostas nesta tese.
Palavras-chave: Elementos Finitos de alta ordem conservativos, Fluxo de Darcy, Meios
The research conducted in this PhD is dedicated to introduce a new numerical approxima-tion methodology for second order elliptic boundary value problems. The originality consist of optimal mass conservation properties. A rigorous new numerical analysis is presented to put the procedure on a solid mathematical framework. This new methodology is based on the Ritz-Galerkin method and allows locally conservative solution of second-order elliptic boundary value problems. For example, solution to fluid dynamics in porous media using high order piecewise polynomial function spaces or multiscale spaces. The solution is simply imposed to satisfy a certain (local) mass conservation restriction on selected subdomains. Alternatively, this requirement can be understood as a generalization of the Finite Volume Method to general functional spaces that allow high order conservative approximations. Based on this new methodology, we introduced a new nonconforming high order conservative finite element method (HOCFEM) for second order elliptic boundary value problems. In parallel, we present some partial results for a modification of the Generalized Multiscale Finite Element Method (GMsFEM) that has a close connection with the methodology addressed in the context of porous media flow. Specifically, we show that the introduced modification improves the efficiency of the method in some cases. Together, HOCFEM and GMsFEM would allow the construction of an efficient and conservative multiscale solver for second-order elliptic boundary value problems. The new methodology is naturally parallelizable, although this advantage is not explored in this the-sis. In addition, preliminary numerical results were obtained by combining HOCFEM with a new Lagrangian-Eulerian method for hyperbolic problems in order to numerically solve and study the nonlinear fundamental problem of incompressible and immiscible water-oil saturation-pressure-velocity flow. Representative numerical experiments are presented and discussed in order to demonstrate the viability of the new methodologies proposed in this thesis.
Keywords: Conservative high-order FEM, Darcy flow, Porous media, High contrast
1 INTRODUCTION . . . . 9
1.1 Motivation . . . 9
1.2 Objectives and Project Proposal . . . 14
1.2.1 Main results and publications. . . 14
1.3 Overview of the Thesis . . . 16
2 HIGH ORDER CONSERVATIVE FINITE ELEMENTS. . . 17
2.1 Introduction . . . 17
2.2 A Review of High Order Finite Volume methods . . . 19
2.3 Brief Review of Multiscale Finite Elements Methods (MsFEM) . . . 23
2.4 Lagrange Multipliers and mass conservation . . . 25
2.5 Conservative Finite Elements Discretization . . . 28
2.6 Analysis . . . 31
2.7 The Case of Piecewise Polynomials of Degree r in Regular Square Meshes . . . 35
2.8 The Case of Highly Anisotropic Medium . . . 40
3 NUMERICAL EXPERIMENTS . . . 42
3.1 Conservative FEM in Scalar Homogeneous Medium . . . 42
3.1.1 Smooth right hand side and Dirichlet’s boundary conditions. . . 42
3.1.2 Problems with Neumann boundary condition . . . 47
3.1.2.1 Smooth right hand side . . . 48
3.1.2.2 Singular right hand side . . . 49
3.2 HOCFEM and Anisotropic Coefficient. . . 51
3.3 HOCFEM in Heterogeneous Medium . . . 53
4 SOME PARTIAL RESULTS AND PERSPECTIVES . . . 58
4.1 Generalized Multiscale Finite Element Method (GMsFEM) . . . 59
4.1.1 Some numerical results for GMsFEM . . . 60
4.2 HOCFEM coupled with a hyperbolic-transport two-phase flow prob-lem . . . 63
5 CONCLUDING REMARKS . . . 70
BIBLIOGRAPHY . . . 72
1 Introduction
The research carried out throughout this PhD is devoted to provide rigo-rous mathematical analysis supported by numerical evidence for a recent methodology introduced in [134, 8]. This new methodology allows for conservative Ritz-Galerkin a-pproximated solutions of second order elliptic boundary value problems associated to fluid dynamics in porous media on piecewise high order polynomial spaces or multiscale spaces. The solution is asked to satisfy some given mass conservation restriction on certain subdomains. This can be understood as an extension of the Finite Volume Method (FV) to general approximation spaces that allows for conservative high order approximations.
In particular, this conservative discretization can be used in conjunction with a recently introduced Generalized Multiscale Finite Element Method (GMsFEM; see [134]). An important part of our research was devoted to a new convergence analysis of GMsFEM [85] which included some modification of the method aiming to improve its performance in the presence of discontinuous and high contrast heterogeneity of the porous media. Although we present only the main results of this specific part of our research, it is necessary to mention that both works are intimately related; the GMsFEM is not conservative so this FV-based methodology was developed to be combined with GMsFEM to obtain a conservative multiscale method in the spirit of [134].
We introduce a new High Order Conservative Finite Element Method (HOCFEM) for second order elliptic boundary value problems based in the methodology introduced in [134] in order to obtain conservative solutions on Finite Element spaces of piecewise high order polynomials. This is a contribution to both areas, Finite Element Methods and Finite Volumes Methods, bringing together too important features of this two worlds, high order approximations and conservation properties.
1.1
Motivation
Transport problems in fluid flow in porous media are modeled by nonlinear partial differential equations (PDEs). The mathematical comprehension of the transport phenomena in multiphase fluid flow dynamics in porous media is very important in many contexts, from pure mathematics (uniqueness, existence and regularity of solutions) to numerical analysis [6,5,14,79,52,110,115,141,102] and applications [57,4,3,6]. There are many challenges and difficulties in the mathematical modeling and numerical analysis of the PDEs that govern multiphase fluid flow in porous media (see, e.g., [102,57, 101]). One difficulty arises from the inherent multiscale nature of the problem induced by the geological properties of the rocks that compose the reservoir (e.g.,permeability and porosity)
modelled as highly variable and discontinuous coefficients [104]. Also, analytic solutions are very difficult (e.g., [57, 104, 84,52,79, 122, 110]) and sometimes impossible to obtain with traditional techniques due to the nonlinear nature of these PDEs. This means that numerical analysis of algorithms is an alternative in the mathematical understanding of these PDEs.
Let Ω P R2 (or R3) be a bounded domain and J “s0, T r be a time interval. Consider the two-phase flow equation for incompressible and immiscible flow, with water and oil phases, in phase formulation [48, 3],
B
BtpφpxqSwq ` divpvpSwqfwpSwqq “ ∇ ¨ ww x P Ω, t P J, (1.1a) div pvq “ 0, v “ ´Kpxqλ pSwq ∇po` uwo x P Ω t P J, (1.1b)
where the water saturation (Sw), the Darcy’s flux (v) and the oil pressure (po) are the
unknown and its dependence on x and t is omitted in the equations in order to simplify the notation. The term φpxq is the porosity, and Kpxq is the absolute permeability tensor. Phase relative mobility λipSwq, total mobility λpSwq and fractional flow function fipSwq
are λipSwq “ kipSwq µi , λpSwq “ ÿ i λipSwq, fipSwq “ λipSwq λpSwq , i “ o, w, (1.2) where ki is the relative permeability and µi the viscosity in each phase. The diffusive flux
ww and the correction term uwo are given by
ww “ ´Kpxq rλwpSwqp1 ´ fwqs ∇pc; uwo “ ´KpxqλwpSwq∇pc (1.3)
and pc is the capillary pressure, defined as the pressure difference across the interface
between two immiscible fluids arising from the capillary forces in porous medium. If capillary effects are neglected then both uwo and ww vanish. There are several strategies to
approximate the governing equations resulting from basic conservation laws, supplemented by constitutive relations that are typically obtained by laboratory experiments [10, 11,9,
3,49, 48] .
The equations in (1.1b) are known in the literature as the mixed formulation of the pressure equation. Note that we can substitute the expression of v (right term of (1.1b)) in the left term to obtain a second order elliptic equation to be solved for the pressure. Since only one the phase is present in the model, we can get rid of the index and rename (S :“ Sw) and (p :“ po).
Operator splitting techniques aim at the approximation of differential operators (systems of partial differential equations as in (1.1a)-(1.1b)) have been frequently used in problems of simulation of fluid flows in fluid dynamics problems in porous media. Physics based operator splitting separates the underlying phenomena and show clearly its mathematical nature, that is, in the form of hyperbolic, parabolic, elliptic or mixed models
in the context of (local) classification of PDEs. In this way, different and appropriate numerical techniques can be employed within the operator splitting formulation to provide efficient numerical procedures. We mention some relevant work in the area of fluid dynamics as well as references cited therein [80, 103, 82, 123, 104, 84, 19, 63, 58, 113, 59]. There are several methodologies in the literature for the approximation of PDEs. Basically, this choice must be guided by the mathematical nature of the differential model, given that it is, expected to recover at the discrete level, the same physical and mathematical properties of the continuum model.
In this thesis, we employ a splitting operator technique where the saturation from the previous time step is used to update the mobility coefficient required by the pressure solver and subsequent velocity calculation (see, e.g., [62]). Once this velocity is available, it is used in conjunction with an explicit saturation time-marching scheme for a specified number of time steps. The updated saturation is used again in order to update the pressure, and the process is continued until a final simulation time is reached. See Figure 1for a schematic of the operator splitting.
Figure 1 – An illustration of the decomposition of operators technique used to solve the two-phase model problem
The previous discussion motivates the identification of a hyperbolic (transport) problem (1.4) and a Poisson’s (pressure) problem (1.5) to be sequentially solved.
Bpφ Spx, tqq
Bt ` divpvf pSpx, tqqq “ 0, x P Ω, t P J, Spx, 0q “ ηpxq, x P Ω,
(1.4)
(where ηpxq is the initial condition), and a Poison’s problem associated to elliptic equation ´divpKpxqλpSq∇ppxqq “ qpxq, x P Ω,
ppxq “ pDpxq, x P BΩD,
´KpxqλpSq∇ppxq ¨ npxq “ pNpxq, x P BΩzBΩD,
(1.5)
where Ω P R2 with Lipschitz boundary BΩ, with BΩD being a part of the boundary where
Dirichlet conditions are imposed while Neumann boundary conditions are imposed on BΩzBΩD; let n denote the outer unit normal vector for BΩ.
The focus of this work is to solve the second-order elliptic boundary value problem using the HOCFEM method to be introduced in this thesis and published in [12]. This method is based on the Ritz formulation and incorporates finite volume constraints to ensure local mass conservation using Lagrange multipliers. This methodology will be used (not in this work but in the future) to make the GMsFEM method conservative (see article C. Diaz 2019 [85] in collaboration with E. Abreu and J. Galvis). Putting all together, we justify the development of the HOCFEM method with conservation properties and high order for a precise and efficient numerical approximation of the Poisson model. Given the potential of the new GMsFEM method [85] when combined with conservation properties [7, 12], We also briefly describe some important aspects of the GMsFEM class in light of other discretization methodologies for elliptic problems.
The efficient numerical solution of the second order elliptic problem (1.5) is a very challenging task because of the complex porous media environment and the intricate properties of fluid phases. We can cite several classic techniques for discretization of elliptic problems, such as, Finite Differences [143, 138], Finite Volumes [64, 146], Mimetic finite differences [119, 3, 28,35, 138], Finite Elements (Classical) [34], Discontinuous Galerkin Finite Elements [60,23,132], Mixed Hybrid Finite Elements in Hpdivq spaces [135,137,36] among others, like generalized finite elements [25]. The idea is to obtain a linear system with good mathematical properties, such as “well conditioning” and a local conservative nature at the discrete level, corresponding to the PDE differential operator. It should be noted that all of the above methods require, after discretization of model (1.5), the resolution of a typically poorly conditioned linear algebra problem.
The resolution of large sparse systems of linear equations of the form Ax “ b is key to many numerical simulations in science and engineering problems and is often the most time consuming part of a calculation. The large size of linear systems is due to the discretization (and linearization) of the elliptic PDEs. Even in two-dimensional problems, due to the heterogeneous nature of the porous medium, it is natural to incorporate a notion of uncertainty, which will lead to stochastic PDEs [93,94, 92, 24, 84, 124, 3] and a Monte Carlo approach is a viable alternative. Essentially, this means calculating statistical moments [93, 94, 92, 84] coming from a set of deterministic experiments in orders of hundreds or thousands (desirable) of computational experiments. In order to solve large linear systems it is common to use iterative methods. Still, in some applications iterative methods typically fail and preconditioning is indispensable. Unfortunately, such a remedy is not always sufficient to make it competitive, specially for huge problems where there are billions of unknowns. As a new ingredient, parallel computing has penetrated the same areas of application. As the computer became widely available a new incentive was created to develop methods that take advantage of these powerful computational tools. In general, it can not yet be asserted that an arbitrary sparse linear system can be solved iteratively in an efficient way [147]. For example, if the physical information about the problem in
question can be explored, the most effective and robust methods can be further adapted to obtain these solutions. In fact, this strategy is exploited by multigrid methods, which typically take advantage of multiscale aspects of the model. In addition, parallel machines may require different architectures in such a way that the algorithms require different programming paradigms which are radically different from the classic and consolidated methods in the literature. As a bibliographical review, it is observed that there are several texts on the subjects of multigrid and multiscale, as for example [95, 125,37,130,75,111]. Although a number of other computational packages are available and ready for immediate use, such as [129,128, 127, 126, 140, 37, 95, 125,130]. Even so, high quality open source code requires a considerable amount of time on a not so gently learning curve. For more details on iterative methods with multiscale and multigrid preconditioning, with different approaches and a wide variety of applications, see [140,37, 95, 125, 130].
The main goal of GMsFEM is to construct coarse spaces for Multiscale Finite Element Methods (MsFEM) that result in accurate coarse-scale solutions. This methodology was first developed in [69, 73, 71] based on some previous works [86, 87, 70, 68, 66]. A main ingredient in the construction is the use of an approximation of local eigenvectors (of carefully selected local eigenvalues problem) to construct the coarse spaces. Instead of using one coarse function per coarse node as in classical MsFEM, in the GMsFEM it was proposed to use several multiscale basis functions per coarse node. These basis functions represent important features of the solution within a coarse-grid block and they are computed using eigenvectors of an eigenvalue problem. For applications to high-contrast problems, methodologies to keep small the dimension of the resulting coarse space were successfully proposed [87]. In that paper, the authors made use of coarse spaces that somehow incorporate important modes of a (local) energy related to the problem motivating the general version of the GMsFEM.
The GMsFEM does not guarantee mass conservation properties. In [134] it was designed a mass conservative GMsFEM method. In particular, mass conservation constraints were imposed over control volumes by using Lagrange multipliers and it was numerically validated that the convergence properties of the GMsFEM are maintained while the approximated solution simultaneously satisfies the required conservation properties. Some successful applications of GMsFEM to two-phase problems were presented in [39]. In that work, the authors developed a technique in which GMsFEM solutions could be post-processed to yield mass conservative fluxes for use in two-phase modeling. While the motivation of [39] is similar, the technique presented in [134] yields a system that requires no post-processing, and a solution that automatically yields conservation. In particular, in the present work no additional equations must be solved after obtaining a global solution in order to ensure coarse-grid conservation (in contrast to [39]). Rather, the conservative solution is directly obtained through a single solution of a (slightly larger) global system that has the mass conservation properties automatically embedded. As a
result, no additional computational resources must be allocated for post-processing, and issues such as ill-conditioning of the localized systems may be circumvented. Additionally, the proposed method ensures that the solution obtained through the method proposed in that work is the best among all functions (in the Ritz sense) satisfying the mass conservation restrictions.
1.2
Objectives and Project Proposal
The main goal of our work is to provide a theoretical background for the novel methodology introduced in [134]. This includes convergence proofs and error estimates under classical Finite Element theory assumption. Under classical FE assumptions on meshes, our convergence results holds which means an improvement in the theory of high-order FV methods where analysis is very specific for each model and results usually depend on additional assumptions on mesh parameters. Also, imposing conservation as a restriction, allows to separate the analysis in order to use Finite Element and Finite Volumes classical techniques in the analysis.
We validate our theoretical results by a set of numerical experiments that cover problems restricted to assumptions needed to prove theorems presented, such as, regularity of the solution and homogeneous coefficients. But we also present numerical evidence of the performance of HOCFEM on problems with less regularity or problems with heterogeneous coefficient which are a bit more representative cases in flow in porous media problems.
Without making emphasis on the theory developed in our recent work [8], we want to present some results related to GMsFEM that has an intimate connection with the methodology addressed in this thesis in the context of flow in porous media. The presented methodology allows the construction of an efficient conservative multiscale solver for second order elliptic boundary value problems.
1.2.1
Main results and publications
We list the achievements of this work on a theoretical analysis and a numerical study of HOCFEM by enumerating the specific results obtained as well as the papers we produce and the most important talks we have done.
Main results
• We developed a novel high-order conservative finite element method for second order elliptic problems by using the Ritz formulation of the second order elliptic PDE with conservative constraints imposed via Lagrange multipliers. We obtain locally
high-order approximated solutions of problem (1.5) that yield locally conservative fluxes with no need of post-processing.
• We present a rigorous numerical analysis for the new formulation when applied to the elliptic problem with scalar homogeneous coefficient which includes convergence proofs and error estimates. These theoretical results need no assumptions in meshes other than those from classical theory of finite elements which means a significant improvement in the theory of high order FV Methods.
• We present numerical evidence that, for different configurations of the anisotropy, the resulting linear system is suitable to be solved by using a stationary Richardson’s iteration where the application of the preconditioner corresponds to an approximation of a low-order finite element problem.
• We present numerical evidence that conservation properties also hold for high contrast heterogeneous coefficient.
• Preliminary numerical results were obtain by combining HOCFEM with the Lagrange-an-Eulerian method developed by E. Abreu and J. Perez [15,16,17,13] for hyperbolic problems in order to solve the full set of nonlinear differential equations conforming the saturation-pressure-velocity model.
• It was numerically shown that a modification we made to the generalized multiscale finite element method (GMsFEM) improves its performance in some cases. This is linked to the present work in the sense that the GMsFEM efficiently solves the elliptic problem. Since GMsFEM is not a conservative method, conservation will be imposed using the present methodology in order to develop an algorithm in the spirit of [134].
Papers
• (2018) The main theoretical results for HOCFEM were first published under the title “On high-order conservative finite element methods”[12], in the Journal Computers & mathematics with applications.
• (2019) A modification of GMsFEM, presented in this thesis as a partial result, was published under the title “A convergence analysis of Generalized Multiscale
Finite Element Methods”[7], in the Journal of Computational Physics.
Conference Papers
• (2018) A work to experimentally investigate the performance of HOCFEM in the presence of anisotropic coefficient was published under the name “On high-order
approximation and stability with conservative properties” [85], in the annals of congress: Domain Decomposition Methods in Science and Engineering XIV. Talks
• (2018) A talk on HOCFEM was presented under the name “A high order
conser-vative finite element formulation for Darcy flow problem” at ICM2018 in
the category of Short Communications.
• (2019) A work to couple HOCFEM to a transport problem is in progress. Partial results were presented at InterPore2019 organized in Valencia (6-10 May 2019) under the title “A high-order conservative finite element method for Darcy flow
problem with transport”[8] .
1.3
Overview of the Thesis
The rest of this Thesis is organized as follows. Chapter2is devoted to theoretical analysis of HOCFEM. After a brief introduction in Section 2.1 it follows a quick review of high order Finite Volumes in Section 2.2 and a quick review of multiscale finite elements in Section 2.3. We formally present the methodology to impose mass conservation via Lagrange multipliers in Section 2.4. The construction of HOCFEM using a Ritz-Galerkin approach is introduced in Section 2.5. In Section2.7 we present some theoretical results for the particular choice Qr finite element spaces in regular square meshes. Chapter 3
contains a selection of numerical experiments to verify the theoretical results obtained in previous sections which covers homogeneous case, Dirichlet and Neumann boundary conditions, discontinuous forcing and high contrast heterogeneous medium. In Chapter 4
we present some results related to two researches projects in progress as a perspective for future work: GMsFEM in Section 4.1 and coupling the HOCFEM elliptic solver with a Hyperbolic solver in Section 4.2. Finally, some concluding remarks in Chapter 5.
2 High Order Conservative Finite Elements
This chapter is devoted to provide some general theoretical results for the methodology introduced in [134]. This methodology is developed in order to impose mass conservation on the Ritz formulation of second order elliptic problems. The mass conservation becomes restrictions imposed by using Lagrange multipliers. Related numerical methods arise from a Galerkin-type discretization where several finite dimensional a-pproximation spaces might be consider. If high-order finite element spaces are chosen we call the obtained numerical method: High Order Conservative Finite Element Method (HOCFEM).
2.1
Introduction
A classical and widely used conservative discretization is the Finite Volume (FV) method which provides and approximation of the solution in the space of piecewise linear functions with respect to a triangulation while satisfying mass conservation on elements of a dual triangulation. When higher order discretizations are needed piecewise linear approximation are replaced by other spaces which often means to sacrifice conservation properties.
Finite volumes methods that use higher degree piecewise polynomials have been introduced in the literature previously. The fact that the dimension of the approximation spaces is larger than the number of restrictions led the researchers of [43,50] to introduce additional control volumes to match the number of restrictions to the number of unknowns. An alternative approach is to consider a Petrov–Galerkin formulation with additional test functions rather than only piecewise constant functions on the dual grid. It is important to observe that the additional control volumes require additional computational work to be constructed and in some cases are not easy to construct (see also [42,133]). Additionally, it is well known that piecewise smooth approximations spaces do not perform well for multiscale high-contrast problems [69, 73, 71,86, 70, 68, 66]. Another technique that one can use to obtain a conservative method with richer approximations spaces is the following. In the discrete linear system obtained by a finite element discretization, it is possible to substitute an appropriate number of equations by finite volume equations involving only the standard dual grid. This approach has the advantage that no additional control volume needs to be constructed (see for instance [56] where the authors use a similar approach). It has the flexibility of both FV and FE methods. Some previous numerical experiments suggest a drawback of this approach as the resulting discrete problems may be ill-conditioned for large dimension coarse spaces, especially for higher order approximation
spaces and multiscale problems [56].
In [134] it was proposed the alternative of using a Ritz formulation and cons-tructing a continuous Galerkin-type solution procedure that concurrently satisfies mass conservation restrictions. The resulting linear system of the Galerkin formulation was augmented in the higher order approximation space to impose the finite volume restrictions using a scalar Lagrange multiplier for each restriction. This approach can be equivalently viewed as a constraint minimization problem. The energy functional is minimized restricted to the subspace of functions satisfying the mass conservation on control volumes. Then, in the Ritz sense, the obtained solution is the best among all functions that satisfy the mass conservation restriction. As a main application of the presented techniques , the authors consider the case where the coefficient Λ has high variation and discontinuities (not necessarily aligned with the coarse grid). For this problem it is known that higher order approximation is needed. Indeed, in some cases robust approximation properties, indepen-dent of the contrast, are required. See for instance [73, 71, 86] where it is demonstrated that classical multiscale methods [75] do not render robust approximation properties in terms of the contrast. It is shown that one basis functions per coarse node (with the usual support) is not enough to construct adequate coarse spaces [71, 131]. A similar issue can be expected for the multiscale finite volume method developed in [121, 114, 108] and related works, when applied to high-contrast multiscale problems since the approximation spaces have similar approximation properties. In the case of Galerkin formulations, robust approximation properties are obtained by using the Generalized Multiscale Finite Element Method (GMsFEM) framework.
Finally we mention the works on MsFV. When dealing with multiscale problems, the lack of accuracy in the use of classical spaces led researchers to introduce enriched spaces (as well as iterative procedures). In [56] the authors construct an enrichment of the initial coarse space spanned by nodal basis functions. We also note that a hybrid finite volume/Galerkin formulation for the coarse-scale problem is devised. The resulting method has all features of the MsFV method, but is more robust and shows improved convergence properties if used in an iterative procedure. In comparison, in GMsFEM framework the additional degrees of freedom are related to local eigenvalue problems. This construction is related to the approximation properties the resulting space will have and is motivated by the numerical analysis of the interpolation operator into the coarse space. Furthermore, instead of a hybrid finite volume Galerkin formulation, it is employed a Ritz formulation in the space of functions satisfying the mass conservation.
2.2
A Review of High Order Finite Volume methods
As it was pointed out in Chapter 1, a central part of this work is motivated by the relevance of obtaining high order conservative numerical methods for boundary value problems associated with second order elliptic PDEs. This is a fundamental problem when solving the equations that model fluid flows in porous media. The evolution of methods based on Finite Volumes yields important challenges to be faced.
The low order of approximation of FV leads to slow solvers. To have a fast solver we can use high order Finite Element Methods (FEM) but they happen to be not-locally conservative. To address this issue there are alternatives such as performing a post-processing of the FEM-solution, changing the discretization to a FV-based method, using a Mixed formulation, using discontinuous formulations and others.
Let’s see some examples. The classical FV method consists in imposing con-servation by integrating the equation (1.5) over local control volumes and then applying Gauss divergence theorem to obtain
ż BV vh¨ n “ ż V q, (2.1)
where V runs over a set of control volumes and vh¨ n is the computed approximation of
the flux. Clearly, by using this formulation there will be no need of post-processing and calculation of conservative fluxes is straight forward.
So let Th and Th˚ be primal and dual meshes on Ω where each element in
T˚
h contains a single node of Th. Let Q1pThq be the space of continuous piecewise linear
functions on Th and denote tVjuj“1¨¨¨N the elements of Th˚. Our problem consists of
computing p P Q1pThq such that
ż BVj ´Λ∇p ¨ n “ ż Vj q, @ Vj P Th˚, (2.2)
This is equivalent to solve
ˆ A0u “ ¯b0, (2.3) here ¯A0 “ r¯aijs with ¯aij “ ´ ż BVj Λ∇φi ¨ n and ¯b0 “ r¯bjs “ ż Vj q. Where tφiui“1¨¨¨N is
the usual basis of Q1pThq .
To overcome the low order rate of convergence of FV, many mathematicians have worked to modify it to obtain high order variants. The new methods obtained are called Finite Volume Element method (FVE). New challenges arise from this formulation such as finding a systematic way to construct FVE spaces and the development of a general theory of approximation as it exists for FEM and finite difference method.
A FVE method is designed by mean of the Petrov-Galerkin formulation in which the solution space (Sh) and the test space (Vh) are no longer the same as it used to
be FEM. To design a higher order FVE method, we will partition the space in two different meshes, a mesh of elements (Th) associated to Sh and a mesh of control volumes (Th˚)
associated to (Vh). The first requirement is that the number of degrees of freedom of the
test space Vh and the solution space Sh should be the same, and the second requirement
is that the test space should include the characteristic functions of the control volumes so that local conservation will be preserved. Polynomials remain the typical option for basis in both spaces so we can name a FVE scheme by the order of its polynomial bases. For example, if Sh and Vh has a piecewise polynomial basis of order k and k1 we call the
scheme of k ´ k1 FVE.
Approximation theory for higher order FVE method differs considerably from the corresponding FEM. Non-conforming and non-symmetric discretizations result in difficulties for establishing a general analysis framework. The main idea for analyzing higher order FVE methods is to first derive the elementary matrix forms, and then manage to obtain positiveness and boundedness of the resulted matrices under certain geometric assumptions.
In one dimension, higher-order FVE methods for two-point boundary value problem was introduced in 1982 in [117] for the linear case (1D). The solution space Sh
use usual finite elements on Th of a domain pa, bq. The dual partition Th˚ was built by
the midpoints of the primal elements. The basis functions tΦrjuj“1¨¨¨N for the test space
Vh were chosen from a set of piecewise Taylor-polynomials of grade r while the basis
functions tφrjuj“1¨¨¨N for the solution space Sh vary among usual r ´ order Finite Elements
ones. Optimal order error estimates in the H1-norm for these methods can be derived by investigating directly positiveness and boundedness of the resulted matrices, see, e.g., [46, 117].
The analysis for quadratic (2-0) and cubic (3-0) FVE methods based on Barlow points could be found in [89, 99]. A class of high-order FVE schemes with spectral-like spatial resolution characteristics is developed in [50]. An implicit reduction of the number of unknowns was obtained by a local implicit mapping of some degrees of freedom. Recently, a family of arbitrary order FVM schemes, with control volumes based on the roots Gauss points, were constructed and analyzed in a unified approach [41], the solution space was chosen as the Lagrange finite element with Lobatto’s interpolation points, significantly different from other FVE methods. With help of the inf-sup condition, the optimal order convergence in the H1-and L2-norms were derived.
For FVE methods in higher dimensions, a difficult task is to construct a suitable dual partition to ensure the resolvability of the resulting system of equations. Certain geometric requirements have to be specified for different higher order methods
(see examples of elements and volumes elements in Figure 2).
Figure 2 – Examples of primal and dual meshes where K˚ are control volumes
In [144], authors studied higher-order FVMs on triangular meshes based on Lagrange quadratic elements for the two-dimensional Poisson equation obtaining an optimal H1-error estimate under the assumption that the maximum angle of each element of the triangulation is not greater than π{2 and the ratio of the lengths of the two sides of maximum angle is within the interval rp2{3q1{2, p3{2q1{2s. In [45] authors analyzed the FVM of the 3rd order Hermite type (Hermite 3 ´ 1 FVM) and obtained an optimal H1 error estimate under the same assumption on the meshes described above. In [118] authors developed a FVE scheme in terms of quadratic basis functions (2 ´ 0 FVE) with sufficient conditions on the triangulation which lead to an optimal H1 error estimate. In [149] authors analyzed linear and quadratic FVE schemes for quadratic FVM on two dimensional domains, they proved that the inf-sup condition holds if θ0 ě 20.95˝p« 0.16639πq for FVM scheme
proposed by [144], if θ0 ě 7.11˝p« 0.0395πq for FVM scheme proposed by [78], and if
θ0 ě 9.98˝p« 0.05544πq for FVE scheme proposed by [118], where θ0 denotes the minimal
angle of the partition.
Recently [50] provided a systematic study of the geometric requirements for higher order FVE methods on triangular meshes. Necessary and sufficient conditions for the uniformly local ellipticity condition of higher order FVE schemes are introduced.
Quadrilateral meshes can be regarded as mappings from rectangular ones. Measuring the effect of the distortion becomes a critical task. For an affine bi-quadratic FVE method, the primal partition Th could be a conforming quadrilateral mesh, whereas
the dual partition T˚
h can be constructed as shown in Figure 3: each edge of Q P Th is
partitioned into three segments so that with length relation 1 : n : 1. We connect these partition points with line segments to the corresponding points on the opposite edge. This way, each quadrilateral of Th is divided into nine sub-quadrilaterals Qz, z P ZhpQq, where
ZhpQq is the set of the vertices, the midpoints of edges, and the center of Q. For each
Figure 3 – Quadrilateral Primal and dual meshes
subregions Qz containing the node z. Therefore, we obtain a collection of control volumes
covering the domain Ω. This is the dual partition T˚
h of the primal partition Th. The
solution space Sh is chosen as the affine biquadratic Lagrange elements on Th, while the
test space S˚
h is chosen as the piecewise constants on T
˚
h .
In [150] the authors studied a 2 ´ 0 FVE method with n “ 2. Optimal order H1 error was proved under the almost parallelogram mesh assumption. The method was extended to three-dimensional problems on right quadrangular prism grids [151], where the ratio of the segments for control volumes is 1 : 4 : 1 to ensure the symmetry. A few years latter [151] developed a finite volume element method with affine quadratic bases on right quadrangular prism meshes for three-dimensional elliptic boundary value problems. The optimal second order H1 error is proved under the almost parallelogram mesh assumption and numerical results were presented to illustrate the theoretical analysis.
It was proposed a high (arbitrary) order FVE method for elliptic problems in [145]. Piecewise polynomial basis functions are used as trial functions while the control volumes are constructed by a vertex-centered technique. The discretization is tested on numerical examples utilizing triangles and quadrilaterals in 2D. They achieved optimal order H1 error while L2 error was one order below optimal for even degrees of polynomials and optimal for odd degrees of polynomial. In [43], L. Chen developed a new class of high order finite volume methods for second order elliptic equations by combining high order finite element methods and linear finite volume methods. He obtain optimal convergence rate in H1-norm over two-dimensional triangular or rectangular grids using two particular geometric construction of the dual mesh.
volu-me volu-methods on rectangular volu-meshes was developed in [50]. The authors establish the boundedness and uniform ellipticity of the bilinear forms for the methods, and show that they lead to an optimal error estimate of the methods. They prove that the uniform local-ellipticity of the family of the bilinear forms ensures its uniform ellipticity. Necessary and sufficient conditions for the uniform local-ellipticity in terms of geometric requirements on the meshes of the domain of the differential equation were established.
FVE schemes of any order over rectangular meshes are presented in [152]. The dual partition was constructed based on Gauss points and they prove optimal rate of convergence in L2pΩq. As a continuation of this work, [153] constructed a family of any order FVE schemes also based on the Gauss points over quadrilateral meshes. Using a novel mapping from the trial to test space, they provide a unified proof for the convergence-order property of the schemes. Moreover, to prove this optimal-convergence-order property, they only required a very relaxed mesh condition which is similar to that in FEM.
2.3
Brief Review of Multiscale Finite Elements Methods (MsFEM)
Multiscale methods have been developed in the last few decades to approximate efficiently problems involving second order elliptic partial differential equations. These problems are very important in several areas of research, in particular in applications to oil reservoir simulation with high contrast in heterogeneity. Despite considerable advances in computational processing and storage, traditional methods that have been used in mainstream oil reservoir simulators are not capable of dealing with problems involving billions of elements in the discretization of large computational regions. Very large reservoirs of interest to the industry can be found, for instance, in the Brazilian pre-salt layer and acceptable accuracy in numerical simulations require a considerable amount of elements. A number of multiscale methods have been developed to overcome the computational challenges posed by these simulations to ensure acceptable precision of numerical solutions. Multiscale methods based on domain decomposition techniques divide the global domain into subregions that may be overlapping or non-overlapping, and such decomposition facilitates the use of parallelization techniques. Local solutions (or multiscale basis functions) are constructed through solutions of families of boundary value problems within each subdomain. These functions retain fine mesh information related to the contrast of the porous medium properties and are employed as building blocks to construct global approximations for the problem at hand. The key idea is to obtain an approximate solution considering unknowns defined on a coarse scale, the skeleton of the underlying domain decomposition, and thus reducing drastically the number of unknowns with respect to thefine mesh.
Multiscale methods can be classified into two major classes: those in the context of finite elements and those that use finite volumes. The works of [108,107,109,116,120] are some references, among many others, for the development and applications of Multiscale Finite Volume Methods in numerical simulations of oil reservoirs. In the context of multiscale mixed methods of particular interest to our developments we mention the Multiscale Mortar Mixed Finite Element Method (MMMFEM) [22,88,148], the Multiscale Hybrid-Mixed Method (MHM) [20, 100] the Multiscale Mixed Method (MuMM) [83, 18] and the Multiscale Robin Coupled Method (MRCM) [97, 98] that has been recently introduced in the literature. The MMMFEM couples subdomains through a continuous pressure and weak continuity of normal fluxes. Thus, a post-processing step is inevitable to produce velocity field with continuous normal components on the fine grid. On the other hand the MHM couples subdomain through the imposition of continuous normal components, and the pressure is weakly continuous.
The design and mathematical analysis of high-contrast multiscale problems continue being a challenging problem; See for instance [105, 1, 74, 31, 2, 21, 22, 44, 74,
107, 54, 106, 75, 87, 38, 55]. These approaches approximate the effects of the fine-scale features using a coarse mesh. They attempt to capture the fine scale effects on a coarse grid via localized basis functions. The main idea of Multiscale Finite Element Methods (MsFEMs) is to construct basis functions that are used to approximate the solution on a coarse grid. The accuracy of MsFEMs is found to be very sensitive to the particularities of the construction of the basis functions (e.g., boundary conditions of local problems, See for instance [75, 76, 77]). It is known that the construction of the basis functions need to be carefully designed in order to obtain accurate coarse-scale approximations of the solution (e.g., [75]). In particular, the resulting basis functions need to have similar oscillatory behavior as the fine-scale solution. In classical multiscale methods, a number of approaches are proposed to construct basis functions, e.g., oversampling techniques or the use of limited global information (e.g., [105,75]) that employs solutions in larger regions to reduce localization errors. Recently, a new and promising methodology was introduced for the construction of basis function. This methodology is referred as to Generalized Multiscale Finite Element Method (GMsFEM). The main goal of GMsFEMs is to construct coarse spaces for MsFEMs that result in accurate coarse-scale solutions. This methodology was first developed in [87, 70] in connections with the robustness of domains decomposition iterative methods for solving the elliptic equation with heterogeneous coefficients subjected to appropriate boundary conditions
A main ingredient in the construction was the use of local generalized eigenvalue problems and (possible multiscale) partition of unity functions to construct the coarse spaces. Besides using one coarse function per coarse node, in the GMsFEM it was proposed
to use several multiscale basis functions per coarse node. These basis functions represent important features of the solution within a coarse-grid block and they are computed using eigenvectors of an eigenvalue problem. Then, in the works [73, 40, 68], some studies of the coarse approximation properties of the GMsFEM were carried out. In these works and for applications to high-contrast problems, methodologies to keep the dimension of the resulting coarse space small were successfully proposed. The use of coarse spaces that somehow incorporates important modes of a (local) energy related to the problem motivated the general version of the GMsFEM. Thus, a more general and practical GMsFEM was then developed in [69] where several (more practical) options to compute important modes to be include in the coarse space was used. See also [72] for an earlier construction. It is important to mention that the methodology in [69] was designed for parametric and nonlinear problems and can be applied for variety of applications as it has been shown in recent developments not reviewed here.
2.4
Lagrange Multipliers and mass conservation
Consider the boundary value problem (1.5) with homogeneous Dirichlet and homogeneous Neumann boundary conditions.
divpΛpxq∇pq “ q, in Ω, p “ 0, on BΩD,
´Λpxq∇p ¨ n “ 0, on BΩzBΩD,
(2.4)
where Ω is a bounded domain in R2 with Lipchitz boundary BΩ and we have define the coefficient Λpxq :“ KpxqλpSq in order to simplify the notation. Denote HD1pΩq the space of functions in H1pΩq which vanish on BΩD. The analysis of problem (1.5) restricted to
homogeneous boundary conditions in BΩD covers the case of non homogeneous conditions
satisfying p “ pD on BΩD where pD P H1{2pBΩDq. By Lemma (.0.3) there exists a function
ˆ
p P H1pΩq such that ˆp|BΩD “ pD, we just need to replace the right hand side in 2.4 by
q ´ divpΛ∇ˆpq, to be interpreted in the weak sense, and obtain an equivalent problem with homogeneous boundary conditions on BΩD. We could also consider pD as a function in
H1pΩq such that p|BΩD “ pD|BΩD on BΩD and do as before to obtain an equivalent problem
with homogeneous boundary conditions on BΩD. A formal interpretation of pD|BΩD can be
found in [34, Chapter 1].
In case |ΩD| “ 0, HD1pΩq is space of functions in H
1
pΩq with zero average on Ω. The variational formulation of problem (1.5) is to find p P HD1pΩq such that
app, vq “ F pvq for all v P HD1pΩq, (2.5) where, for p, v P HD1pΩq, the bilinear form a : HD1pΩq ˆ H
1 DpΩq Ñ R is defined by app, vq “ ż Ω Λpxq∇ppxq∇vpxq dx, (2.6)
for v P HD1pΩq the functional F : HD1pΩq Ñ R is defined by F pvq “
ż
Ω
qpxqvpxq dx. (2.7)
In order to consider a general formulation for porous media applications we let Λ be a 2 ˆ 2 matrix with entries in L8
pΩq to be almost everywhere symmetric positive definite matrix with eigenvalues bounded uniformly from below by a positive constant. In certain parts of the paper when analysis and regularity theory are required, we assume Λ ” I.
The bilinear form a is known to be continuous and coercive under the above conditions on Λ while the functional F is continuous provided q P L2pΩq; therefor, Lax-Milgram theorem guaranties existence and uniqueness of solution of the variational problem (2.5). As it is usual in finite elements we define the seminorm } ¨ }a“
a
ap¨, ¨q. If Dirichlet boundary conditions are imposed, Friedrichs’ inequality (see [34, p. 135]) guaranties that } ¨ }a is in fact a norm and we call it the energy norm.
Symmetry of the bilinear form a implies that problem (2.5) is equivalent to the minimization problem (see [25, p. 58]) : Find p P HD1pΩq such that
p “ arg min vPH1 DpΩq J pvq, (2.8) where J pvq “ 1 2apv, vq ´ F pvq. (2.9)
In order to deal with mass conservation properties we follow the method introduced in [134]. Before continuing with the description of the problem we introduce the meshes we are going to use in our discrete problem. Let Th “ tRjuNj“1h be the partition
of Ω where elements are triangles or squares. Here Nh is the number of elements of the
partition. We also have a dual mesh T˚
h “ tVku N˚
h
k“1 where the elements are called control
volumes and N˚
h is the number of such volumes. In general it is selected one control volume
Vk per vertex of the primal mesh not in BΩD. In case |BΩD| “ 0, Nh˚ is the total number of
vertices of the primal partition including the vertices on BΩ. Figure4illustrates primal and dual meshes defined by of squares where BΩD “ BΩ, in this case Nh˚ is equal to the number
of interior vertices of the primal triangulation. We assume that each control volume Vk
has a Lipchitz continuous boundary BVk.
To ensure mass conservation, we impose it in each control volume tVkuNk“1h as a
restriction. We mention that our formulation allows for a more general case where only few control volumes, not related to the primal mesh, are selected.
Let us define the linear functional τkpvq “
ż
BVk
´Λ∇v ¨ nds, 1 ď k ď Nh˚. We
first note τkpvq is not well defined for v P HD1pΩq. To fix that, recall that q P L
2
therefor, we introduce the Hilbert space Hdiv1 ,ΛpΩq “ v P H 1 DpΩq and Λ∇v P Hpdiv, Ωq( , with norm }v}2H1 div,ΛpΩq “ }v} 2 a ` }divpΛ∇vq} 2
L2pΩq where the divergence is taken in the
weak sense. We note that this space and norm are well-defined with the properties of Λ described above, that is, the smaller eigenvalue of Λpxq is uniformly bounded from below by a positive number, by using similar arguments given in [136] Theorem 1.
Remark 2.4.1. If we integrate by parts
ż
Vk
divpΛ∇vqz with the function z “ 1, we see that τk is now a well defined continuous linear functional on Hdiv1 .ΛpΩq; the integration
by parts can be performed since ż
Vk
Λ∇v ¨ ∇v dx ` }divpΛ∇vq}2L2pV
kq is well-defined and
bounded by }v}2a` }divpΛ∇vq}2L2pΩq [136, pag. 393]. and
Let p be the solution of (2.5) and define mk “ τkppq “
ż
Vk
qds, 1 ď k ď N˚
h .
The problem (2.8) is substituted by: Find p P Hdiv1 ,ΛpΩq such that p “ arg min
vPWJ pvq, (2.10)
where the subset of functions that satisfies the mass conservation restrictions is defined by W “!v P Hdiv1 ,ΛpΩq such that τkpvq “ mk, 1 ď k ď Nh˚
)
Ď HD1pΩq.
Problem (2.10) above can be viewed as Lagrange multipliers min–max opti-mization problem (see [29] and references therein). Then, in case an approximation ph of p, is required to satisfy the constraints τkpphq “ mk, 1 ď k ď Nh˚ , we can do that by
discretizing directly the formulation (2.10). In particular, we can apply this approach to a set of mass conservation restrictions used in finite volume discretizations.
In order to proceed with the associated Lagrange formulation we define Mh “ Q0DpT
˚
hq “ tµ P Q
0
pTh˚q : µ|V “ 0 @ V P Th˚ : BV X BΩD ‰ ∅u. (2.11)
Therefor, Mh is the space of piecewise constant functions on the dual mesh T˚
h that
vanish on each V that intersects BΩD non trivially. The Lagrange multiplier formulation
of problem (2.10) can be written as (see, [91, Theorem 4.2]): Find p P Hdiv1 ,ΛpΩq and λ P Mh that solve,
pp, λq “ arg max
µPMhvPHmin1
div,ΛpΩq
J pvq ` papv, µq ´ ¯F pµqq. (2.12) Here, the total flux bilinear form a : Hdiv1
,ΛpΩq ˆ M h Ñ R is defined by apv, µq “ N˚ h ÿ k“1 µk ż BVk
The functional ¯F : Mh Ñ R is defined by ¯ F pµq “ Nh ÿ k“1 µk ż Vk q for all µ P Mh.
Note that problem (2.12) depends on Th˚ and therefor depends on h. The first order conditions of the min-max problem above give the following saddle point problem: Find p with p P Hdiv1 ,ΛpΩq, and λ P Mh that solve,
app, vq ` apv, λq “ F pvq for all v P Hdiv1 ,ΛpΩq,
app, µq “ ¯F pµq for all µ P Mh. (2.14)
See for instance [29]. Note that if the exact solution of problem (2.8) satisfies the restrictions in the saddle point formulation above, then we have λ “ 0 and we obtain the uncoupled system
app, vq “ F pvq for all v P Hdiv1 ,ΛpΩq,
app, µq “ ¯F pµq for all µ P Mh. (2.15)
Also observe that the second equation above corresponds to a family of equations, one for each h, all of them being satisfied by p.
2.5
Conservative Finite Elements Discretization
Recall that we have introduced a primal mesh Th “ tRjuNj“1h defined by elements
that are triangles or squares. Here Nh is the number of elements of the partition. We
also have given a dual mesh T˚
h “ tVku N˚
h
k“1 where the elements are called control volumes.
Figure 4illustrate a primal and dual mesh defined by squares.
Let us consider Ph “ QrpThq the space of piecewise continuous polynomial
functions of degree r on each element of the primal mesh, and PDh “ Ph X HD1pΩq
(which are the functions in Ph that vanish in BΩD). A basis of PDh will be the standard
basis of Ph without those functions associated to nodes on BΩD. Since QrpThq Ć Hdiv,Λ1
we are considering a nonconforming method but the theoretical analysis carried out is different from the classical analysis for nonconforming methods and we remark that the particularities of nonconforming methods are addressed in our analysis but in a different way.
Let Mh “ Q0pTh˚q be the space of piecewise constant functions on the dual
mesh T˚
h. We mention here that our analysis may be extended to different spaces and
differential equations. See for instance [134] where the authors consider GMsFEM spaces instead of piecewise polynomials.
The discrete version of (2.14) is to find ph P PDh and λ P M h
such that apph, vhq ` apvh, λhq “ F pvhq for all vh P PDh,
apph, µhq “ ¯F pµhq for all µh P Mh.
Figure 4 – Example of regular mesh defined by squares and its dual partition.
Let tϕiu be the basis of PDh. We define the matrix
A “ rai,js where aij “
ż
Ω
Λ∇ϕi¨ ∇ϕj. (2.17)
Note that A is the finite element stiffness matrix corresponding to finite element space PDh. Introduce also the restriction matrix
¯
A “ rak,js where aij “
ż
BVk
Λ∇ϕj¨ n. (2.18)
With this notation, the matrix form of the discrete saddle point problem is given by, AUh “ « A A¯T ¯ A O ff « uh λh ff “ « f ¯ f ff , (2.19)
where vectors f and g are defined by, f “ rfksNk“1h with fk “ ż Ω q ¨ ϕk, f “ r ¯¯ fks N˚ h k“1 with f¯k “ ż Vk q. (2.20) For instance, in the case of the primal and dual triangulation of Figure 4 and r “ 2, the finite element matrix A is a sparse matrix with a principal diagonal and 8 superior and inferior sub-diagonals. Also, a control volumes Vk has not empty interception with, at
most, 9 supports of basis functions ϕj, see Figure5. Then, matrix ¯A is also sparse.
Remark 2.5.1. Note that matrix ¯A is related to classical (low order) finite volume matrix. Matrix ¯A is a rectangular matrix with more columns than rows. Several previous works on conservative high-order approximation of second order elliptic problem have been designed by “adding” rows using several constructions. Summarizing, the strategies are:
Figure 5 – Control volumes that intersect the support of a Q2 basis function.
1. Construct additional control volumes and test the approximation spaces against piecewise constant functions over the total of control volumes (that include the dual grid element plus the additional control volumes). We mention that constructing additional control volumes is not an easy task and might be computationally expensive. We refer the interested reader to [47, 50, 51] for additional details.
2. Use additional basis functions that correspond to nodes other than vertices to obtain an FV/Galerkin formulation. This option has the advantage that no geometrical constructions have to be carried out. On the other hand, this formulation seems difficult to analyze. Also, some preliminary numerical tests suggest that the resulting linear system becomes unstable for higher order approximation spaces (especially for the case of high-contrast multiscale coefficients).
3. Use the Ritz formulation with local mass conservation restrictions (2.16).
Note that if r “ 1 the restriction matrix (A) in the linear system (2.19) corresponds to the usual finite volume matrix. This matrix is known to be invertible [81]. In this case, the affine space W is a singleton. Moreover, the only function u satisfying the restriction is given by u “ ¯A´1f . Then, finding the solution of (2.16) reduces to the
classical finite volume method.
The solution of the associated linear system (2.16), which is a saddle point linear system can be readily implemented using efficient solvers for the matrix A ; See for instance [29]. Additionally, we mention that the analysis of the method can be carried out using usual tools for the analysis of restricted minimization of energy functionals and mixed finite element methods.
2.6
Analysis
We show next that imposing the conservation in control volumes using Lagrange multipliers does not interfere with the optimality of the approximation in the H1 norm. As we will see, imposing constraints results in non optimal L2 approximation but we were able to reformulate the method to get back to the optimal L2 approximation.
Before proceeding we introduce notation to avoid proliferation of constants. We use the notation A ĺ B to indicate that there is a positive constant C1 such that
A ď C1B. If additionally there exists C2 such that B ď C2A we write A — B. These
constants do not depend on u, uh, λh, q, but they might depend on Λ, the shape regularity of the elements and the shape of Ω.
Denote }v}2a “
ż
Ω
Λ∇v ¨ ∇v for all v P HD1pΩq and let us remind that
Hdiv1
,Λ :“ tv P H
1
DpΩq : Λ∇v P Hpdiv, Ωqu, and set V h
“ PDh X H
1
div,Λ.
Assumption A: There exist norms } ¨ }Vh and } ¨ }Mh for Vh and Mh,
respec-tively, such that
1. Augmented norm: }v}aď }v}Vh for all v P Vh.
2. Continuity: there exists }¯a} P R` such that
|¯apv, µhq| ď }¯a}}v}Vh}µh}Mh @v P Vh and µh P Mh. (2.21)
3. Inf-Sup: there exists α ą 0 such that inf µhPMh sup vhPVh apvh, µh q }vh}Vh}µh}Mh ě α ą 0. (2.22)
Remark 2.6.1. The Inf-Sup condition above can be replaced by: there exists α ą 0 such
that inf µhPMh sup vhPVh apvh, µhq }vh}a}µh}Mh ě α ą 0. (2.23)
We present a concrete example of the norms of Vh and Mh in the next section.
Assume that tp, λu is the solution of (2.14) and tph, λhu the solution of (2.16). We have the following result which reminds us of Céa’s Lemma and the proof uses classical approximation techniques for saddle point problems.
Theorem 2.6.2. Assume that “Assumption A” holds. Then
}p ´ ph}a ď 2 ˆ 1 ` }a} α ˙ inf vhPVh}p ´ v h }Vh.
Proof. Note that in both problems, (2.15) and (2.16), µ belongs to the finite dimensional subspace Mh. Also, the exact solution of the Lagrange multiplier component of (2.15) is λ “ 0. Now we derive error estimates following classical saddle point approximation analysis. Define
Whpqq :“ vh P Vh : apvh, µq “ ¯F pµq for all µ P Mh(
and
Wh :“ vh P Vh : apvh, µq “ 0 for all µ P Mh( .
First we prove
}p ´ ph}aď 2 inf
whPWhpqq}p ´ w
h
}a. (2.24)
The inf-sup condition in (2.23) along with continuity of a, a, F , and F complete the hypothesis of Ladyzhenskaya–Babuska–Brezzi condition [36,32] and therefor the saddle point problem (2.16) has a unique solution which implies that Whpqq (as well as Wh) is not empty. Take any wh P Whpqq and solve, for vh P Wh, the problem
apvh, zhq “ F pzhq ´ apwh, zhq for all zh P Wh. (2.25)
Since a is bounded and coercive there exists a unique solution of previous problem. Put ph “ vh` wh, then ph is the unique solution of the problem: find ph P Whpqq such that
apph, zhq “ F pzhq for all zh P Wh
which is equivalent to problem (2.16) (see [91]). We have from (2.15), (2.16) and (2.25) that apvh, vhq “ apph´ wh, vhq “ apph, vhq ´ apwh, vhq “ F pvhq ´ apwh, vhq “ app, vhq ´ apwh, vhq “ app ´ wh, vhq.
Then, by using the ellipticity of a, we have }vh}2a“ apv h, vh q “ app ´ wh, vhq ď }p ´ wh}a}vh}a. (2.26) Then }p ´ ph}aď }p ´ wh}a` }wh´ ph}a ď }p ´ wh}a` }p ´ wh}a “ 2}p ´ wh}a,
