Dans cette thèse, on s’intéresse à de nouveaux schémas de discrétisation afin de résoudre les équations couplées de la poroélasticité et nous présentons des résultats analytiques et numériques concernant des problèmes issus de la poromécanique. Nous proposons de résoudre ces problèmes en utilisant les méthodes Hybrid High-Order (HHO), une nouvelle classe de méthodes de discrétisation polyédriques d’ordre arbitraire. Cette thèse a été conjointement financée par le Bureau de Recherches Géologiques et Minières (BRGM) et le LabEx NUMEV. Le couplage entre l’écoulement souterrain et la déformation géomécanique est un sujet de recherche crucial pour les deux institutions de cofinancement. / In this manuscript we focus on novel discretization schemes for solving the coupled equations of poroelasticity and we present analytical and numerical results for poromechanics problems relevant to geoscience applications. We propose to solve these problems using Hybrid High-Order (HHO) methods, a new class of nonconforming high-order methods supporting general polyhedral meshes. This Ph.D. thesis was conjointly founded by the Bureau de recherches géologiques et minières (BRGM) and LabEx NUMEV. The coupling between subsurface flow and geomechanical deformation is a crucial research topic for both cofunding institutions.

Submitted by Otávio Henrique Perez null (otavio_perez@hotmail.com) on 2017-03-03T20:13:38Z No. of bitstreams: 1 DissertacaoOtavioHenriquePerez.pdf: 2570606 bytes, checksum: dd0f73a1627a83d453f101ef3a973d23 (MD5) / Approved for entry into archive by LUIZA DE MENEZES ROMANETTO (luizamenezes@reitoria.unesp.br) on 2017-03-09T17:45:41Z (GMT) No. of bitstreams: 1 perez_oh_me_sjrp.pdf: 2570606 bytes, checksum: dd0f73a1627a83d453f101ef3a973d23 (MD5) / Made available in DSpace on 2017-03-09T17:45:41Z (GMT). No. of bitstreams: 1 perez_oh_me_sjrp.pdf: 2570606 bytes, checksum: dd0f73a1627a83d453f101ef3a973d23 (MD5) Previous issue date: 2017-02-23 / Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) / Neste trabalho iremos abordar aspectos qualitativos e geométricos a respeito de campos de vetores suaves por partes. Nosso foco será estudar bifurcações locais e globais de codimensão um e dois e também algumas relações de equivalência para campos vetoriais suaves por partes definidos no plano. Classificaremos e caracterizaremos bifurcações genéricas por meio do retrato de fase e do diagrama de bifurcação dos campos envolvidos. Também faremos uma breve introdução sobre Sistemas Slow-Fast. / In this work we study qualitative and geometric aspects of piecewise smooth vector fields. Our focus is to study local and global bifurcations of codimension one and two and some equivalence relations for piecewise smooth vector fields defined on the plane. We will classify and characterize generic bifurcations using the phase portrait and the bifurcation diagram of the vector fields involved. We also incorporate a brief introduction about Slow-Fast Systems. / FAPESP: 2014/18707-6

Le contexte scientifique de cette thèse est l'imagerie sismique dont le but est de reconstituer la structure du sous-sol de la Terre. Comme le forage a un coût assez élevé, l'industrie pétrolière s'intéresse à des méthodes capables de reconstituer les images de la structure terrestre interne avant de le faire. La technique d'imagerie sismique la plus utilisée est la technique de sismique-réflexion qui est basée sur le modèle de l'équation d'ondes. L'imagerie sismique est un problème inverse qui requiert de résoudre un grand nombre de problèmes directs. Dans ce contexte, nous nous intéressons dans cette thèse à la résolution du problème direct en régime harmonique, soit à la résolution des équations d'Helmholtz. L'objectif principal est de proposer et de développer un nouveau type de solveur élément fini (EF) caractérisé par un opérateur discret de taille réduite (comparée à la taille des solveurs déjà existants) sans pour autant altérer la précision de la solution numérique. Nous considérons les méthodes de Galerkine discontinues (DG). Comme les méthodes DG classiques sont plus coûteuses que les méthodes EF continues si l'on considère un même problème à cause d'un grand nombre de degrés de liberté couplés, résultat des approximations discontinues, nous développons une nouvelle classe de méthode DG réduisant ce problème : la méthode DG hybride (HDG). Pour valider l'efficacité de la méthode HDG proposée, nous comparons les résultats obtenus avec ceux obtenus avec une méthode DG basée sur des flux décentrés en 2D. Comme l'industrie pétrolière s'intéresse au traitement de données réelles, nous développons ensuite la méthode HDG pour les équations élastiques d'Helmholtz 3D. / The scientific context of this thesis is seismic imaging which aims at recovering the structure of the earth. As the drilling is expensive, the petroleum industry is interested by methods able to reconstruct images of the internal structures of the earth before the drilling. The most used seismic imaging method in petroleum industry is the seismic-reflection technique which uses a wave equation model. Seismic imaging is an inverse problem which requires to solve a large number of forward problems. In this context, we are interested in this thesis in the modeling part, i.e. the resolution of the forward problem, assuming a time-harmonic regime, leading to the so-called Helmholtz equations. The main objective is to propose and develop a new finite element (FE) type solver characterized by a reduced-size discrete operator (as compared to existing such solvers) without hampering the accuracy of the numerical solution. We consider the family of discontinuous Galerkin (DG) methods. However, as classical DG methods are much more expensive than continuous FE methods when considering steady-like problems, because of an increased number of coupled degrees of freedom as a result of the discontinuity of the approximation, we develop a new form of DG method that specifically address this issue: the hybridizable DG (HDG) method. To validate the efficiency of the proposed HDG method, we compare the results that we obtain with those of a classical upwind flux-based DG method in a 2D framework. Then, as petroleum industry is interested in the treatment of real data, we develop the HDG method for the 3D elastic Helmholtz equations.

Purely numerical methods based on the Finite Element Method (FEM) are becoming increasingly popular in seismic modeling for the propagation of acoustic and elastic waves in geophysical models. These methods o er a better control on the accuracy and more geometrical exibility than the Finite Di erence methods that have been traditionally used for the generation of synthetic seismograms. However, the success of these methods has outpaced their analytic validation. The accuracy of the FEMs used for seismic wave propagation is unknown in most cases and therefore the simulation parameters in numerical experiments are determined by empirical rules. I focus on two methods that are particularly suited for seismic modeling: the Spectral Element Method (SEM) and the Interior-Penalty Discontinuous Galerkin Method (IP-DGM). The goals of this research are to investigate the grid dispersion and stability of SEM and IP-DGM, to implement these methods and to apply them to subsurface models to obtain synthetic seismograms. In order to analyze the grid dispersion and stability, I use the von Neumann method (plane wave analysis) to obtain a generalized eigenvalue problem. I show that the eigenvalues are related to the grid dispersion and that, with certain assumptions, the size of the eigenvalue problem can be reduced from the total number of degrees of freedom to one proportional to the number of degrees of freedom inside one element. The grid dispersion results indicate that SEM of degree greater than 4 is isotropic and has a very low dispersion. Similar dispersion properties are observed for the symmetric formulation of IP-DGM of degree greater than 4 using nodal basis functions. The low dispersion of these methods allows for a sampling ratio of 4 nodes per wavelength to be used. On the other hand, the stability analysis shows that, in the elastic case, the size of the time step required in IP-DGM is approximately 6 times smaller than that of SEM. The results from the analysis are con rmed by numerical experiments performed using an implementation of these methods. The methods are tested using two benchmarks: Lamb's problems and the SEG/EAGE salt dome model. / text

The Boltzmann Transport Equation (BTE) has been the keystone of the kinetic theory, which is at the center of Statistical Mechanics bridging the gap between the atomic structures and the continuum-like behaviors. The existence of solutions has been a great mathematical challenge and still remains elusive. As a grazing limit of the Boltzmann operator, the Fokker-Planck-Landau (FPL) operator is of primary importance for collisional plasmas. We have worked on the following three different projects regarding the most important kinetic models, the BTE and the FPL Equations. (1). A Discontinuous Galerkin Solver for Nonlinear BTE. We propose a deterministic numerical solver based on Discontinuous Galerkin (DG) methods, which has been rarely studied. As the key part, the weak form of the collision operator is approximated within subspaces of piecewise polynomials. To save the tremendous computational cost with increasing order of polynomials and number of mesh nodes, as well as to resolve loss of conservations due to domain truncations, the following combined procedures are applied. First, the collision operator is projected onto a subspace of basis polynomials up to first order. Then, at every time step, a conservation routine is employed to enforce the preservation of desired moments (mass, momentum and/or energy), with only linear complexity. The asymptotic error analysis shows the validity and guarantees the accuracy of these two procedures. We applied the property of ``shifting symmetries" in the weight matrix, which consists in finding a minimal set of basis matrices that can exactly reconstruct the complete family of collision weight matrix. This procedure, together with showing the sparsity of the weight matrix, reduces the computation and storage of the collision matrix from O(N3) down to O(N^2). (2). Spectral Gap for Linearized Boltzmann Operator. Spectral gaps provide information on the relaxation to equilibrium. This is a pioneer field currently unexplored form the computational viewpoint. This work, for the first time, provides numerical evidence on the existence of spectral gaps and corresponding approximate values. The linearized Boltzmann operator is projected onto a Discontinuous Galerkin mesh, resulting in a ``collision matrix". The original spectral gap problem is then approximated by a constrained minimization problem, with objective function the Rayleigh quotient of the "collision matrix" and with constraints the conservation laws. A conservation correction then applies. We also study the convergence of the approximate Rayleigh quotient to the real spectral gap. (3). A Conservative Scheme for Approximating Collisional Plasmas. We have developed a deterministic conservative solver for the inhomogeneous Fokker-Planck-Landau equations coupled with Poisson equations. The original problem is splitted into two subproblems: collisonless Vlasov problem and collisonal homogeneous Fokker-Planck-Landau problem. They are handled with different numerical schemes. The former is approximated using Runge-Kutta Discontinuous Galerkin (RKDG) scheme with a piecewise polynomial basis subspace covering all collision invariants; while the latter is solved by a conservative spectral method. To link the two different computing grids, a special conservation routine is also developed. All the projects are implemented with hybrid MPI and OpenMP. Numerical results and applications are provided. / text

My main research topic is about developing new domain decomposition algorithms for the solution of systems of partial differential equations. This was mainly applied to fluid dynamics problems (as compressible Euler or Stokes equations) and electromagnetics (time-harmonic and time-domain first order system of Maxwell's equations). Since the solution of large linear systems is strongly related to the application of a discretization method, I was also interested in developing and analyzing the application of high order methods (such as Discontinuos Galerkin methods) to Maxwell's equations (sometimes in conjuction with time-discretization schemes in the case of time-domain problems). As an active member of NACHOS pro ject (besides my main afiliation as an assistant professor at University of Nice), I had the opportunity to develop certain directions in my research, by interacting with permanent et non-permanent members (Post-doctoral researchers) or participating to supervision of PhD Students. This is strongly refflected in a part of my scientific contributions so far. This memoir is composed of three parts: the first is about the application of Schwarz methods to fluid dynamics problems; the second about the high order methods for the Maxwell's equations and the last about the domain decomposition algorithms for wave propagation problems.

The Distinct Element Method (DEM) and Discontinuous Deformation Analysis (DDA) are the two most commonly used discrete element methods in rock mechanics. Discrete element approaches are computationally expensive as they involve the interaction of multiple discrete bodies with continuously changing contacts. Therefore, it is very important to ensure that the method selected for the analysis is computationally efficient. In this research, a general assessment of DDA and DEM is performed from a computational efficiency perspective, and relevant enhancements to DDA are developed. The computational speed of DDA is observed to be considerably slower than DEM. In order to identify reasons affecting the computational efficiency of DDA, fundamental aspects of DDA and DEM are compared which suggests that they mainly differ in the contact mechanics, and the time integration scheme used. An in-depth evaluation of these aspects revealed that the openclose iterative procedure used in DDA which exhibits highly nonlinear behavior is one of the main reasons causing DDA to slow down. In order to improve the computational efficiency of DDA, an alternative approach based on a more realistic rock joint behavior is developed in this research. In this approach, contacts are assumed to be deformable, i.e., interpenetrations of the blocks in contact are permitted. This eliminated the computationally expensive open-close iterative procedure adopted in DDA-Shi and enhanced its speed up to four times. In order to consider deformability of the blocks in DDA, several approaches are reported. The hybrid DDA-FEM approach is one of them, although this approach captures the block deformability quite effectively, it becomes computationally expensive for large-scale problems. An alternative simplified uncoupled DDA-FEM approach is developed in this research. The main idea of this approach is to model rigid body movement and the block internal deformation separately. Efficiency and simplicity of this approach lie in keeping the DDA and the FEM algorithms separate and solving FEM equations individually for each block. Based on a number of numerical examples presented in this dissertation, it is concluded that from a computational efficiency standpoint, the implicit solution scheme may not be appropriate for discrete element modelling. Although for quasi-static problems where inertia effects are insignificant, implicit schemes have been successfully used for linear analyses, they do not prove to be advantageous for contact-type problems even in quasi-static mode due to the highly nonlinear behavior of contacts.

We propose a discontinuous Galerkin finite element method (DGFEM) for fully nonlinear elliptic Hamilton--Jacobi--Bellman (HJB) partial differential equations (PDE) of second order with Cordes coefficients. Our analysis shows that the method is both consistent and stable, with arbitrarily high-order convergence rates for sufficiently regular solutions. Error bounds for solutions with minimal regularity show that the method is generally convergent under suitable choices of meshes and polynomial degrees. The method allows for a broad range of hp-refinement strategies on unstructured meshes with varying element sizes and orders of approximation, thus permitting up to exponential convergence rates, even for nonsmooth solutions. Numerical experiments on problems with nonsmooth solutions and strongly anisotropic diffusion coefficients demonstrate the significant gains in accuracy and computational efficiency over existing methods. We then extend the DGFEM for elliptic HJB equations to a space-time DGFEM for parabolic HJB equations. The resulting method is consistent and unconditionally stable for varying time-steps, and we obtain error bounds for both rough and regular solutions, which show that the method is arbitrarily high-order with optimal convergence rates with respect to the mesh size, time-step size, and temporal polynomial degree, and possibly suboptimal by an order and a half in the spatial polynomial degree. Exponential convergence rates under combined hp- and τq-refinement are obtained in numerical experiments on problems with strongly anisotropic diffusion coefficients and early-time singularities. Finally, we show that the combination of a semismooth Newton method with nonoverlapping domain decomposition preconditioners leads to efficient solvers for the discrete nonlinear problems. The semismooth Newton method has a superlinear convergence rate, and performs very effectively in computations. We analyse the spectral bounds of nonoverlapping domain decomposition preconditioners for a model problem, where we establish sharp bounds that are explicit in both the mesh sizes and polynomial degrees. We then go beyond the model problem and show computationally that these algorithms lead to efficient and competitive solvers in practical applications to fully nonlinear HJB equations.

This work is concerned with the introduction of a new higher order numerical scheme based on the discontinuous Galerkin method (DGM). We follow the methodology of higher order finite volume (HOFV) and spectral volume (SV) schemes and introduce a reconstruction operator into the DGM. This operator constructs higher order piecewise polynomial reconstructions from the lower order DGM scheme. We present two variants: the generalization of standard HOFV schemes, already proposed by Dumbser et al. (2008) and the generalization of the SV method introduced by Wang (2002). Theoretical aspects are discussed and numerical experiments with the focus on a 2D advection problem are carried out. Powered by TCPDF (www.tcpdf.org)

Title: Application of hp-adaptive discontinuous Galerkin method to com- pressible flow simulation Author: Karol Tarčák Department: Department of Numerical Mathematics Supervisor: prof. RNDr. Vít Dolejší, Ph.D., DSc. Abstract: In the present work we study an residuum estimate of disconti- nuous Galerkin method for the solution of Navier-Stokes equations. Firstly we summarize the construction of the viscous compressible flow model via Navier-Stokes partial differential equation and discontinuous Galerkin met- hod. Then we propose an extension of an already known residuum estimate for stationary problems to non-stationary problems. We observe the beha- vior of the proposed estimate and modify an existing hp-adaptive algorithm to use our estimate. Finally we apply the modified algorithm on test cases and present adapted meshes from the numerical experiments. Keywords: discontinuous Galerkin method, adaptivity, error estimate 4