Béatrice Rivière

Discontinuous Galerkin Methods For Solving Elliptic And parabolic Equations: Theory and Implementation

Discontinuous Galerkin (DG) methods for solving partial differential equations, developed in the late 1990s, have become popular among computational scientists. This book covers both theory and computation as it focuses on three primal DG methods--the symmetric interior penalty Galerkin, incomplete interior penalty Galerkin, and nonsymmetric interior penalty Galerkin which are variations of interior penalty methods. The author provides the basic tools for analysis and discusses coding issues, including data structure, construction of local matrices, and assembling of the global matrix. Computational examples and applications to important engineering problems are also included. Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation is divided into three parts: Part I focuses on the application of DG methods to second order elliptic problems in one dimension and in higher dimensions. Part II presents the time-dependent parabolic problems without and with convection. Part III contains applications of DG methods to solid mechanics (linear elasticity), fluid dynamics (Stokes and Navier Stokes), and porous media flow (two-phase and miscible displacement). Appendices contain proofs and MATLAB code for one-dimensional problems for elliptic equations and routines written in C that correspond to algorithms for the implementation of DG methods in two or three dimensions. Audience: This book is intended for numerical analysts, computational and applied mathematicians interested in numerical methods for partial differential equations or who study the applications discussed in the book, and engineers who work in fluid dynamics and solid mechanics and want to use DG methods for their numerical results. The book is appropriate for graduate courses in finite element methods, numerical methods for partial differential equations, numerical analysis, and scientific computing. Chapter 1 is suitable for a senior undergraduate class in scientific computing. Contents: List of Figures; List of Tables; List of Algorithms; Preface; Part I: Elliptic Problems; Chapter 1: One-dimensional problem; Chapter 2: Higher dimensional problem; Part II: Parabolic Problems; Chaper 3: Purely parabolic problems; Chapter 4: Parabolic problems with convection; Part III: Applications; Chapter 5: Linear elasticity; Chapter 6: Stokes flow; Chapter 7: Navier-Stokes flow; Chapter 8: Flow in porous media; Appendix A: Quadrature rules; Appendix B: DG codes; Appendix C: An approximation result; Bibliography; Index.

Improved energy estimates for interior penalty, constrained and discontinuous Galerkin methods for elliptic problems. Part I

Three Galerkin methods using discontinuous approximation spaces are introduced to solve elliptic problems. The underlying bilinear form for all three methods is the same and is nonsymmetric. In one case, a penalty is added to the form and in another, a constraint on jumps on each face of the triangulation. All three methods are locally conservative and the third one is not restricted. Optimal a priori hp error estimates are derived for all three procedures.

A Priori Error Estimates for Finite Element Methods Based on Discontinuous Approximation Spaces for Elliptic Problems

We analyze three discontinuous Galerkin approximations for solving elliptic problems in two or three dimensions. In each one, the basic bilinear form is nonsymmetric: the first one has a penalty term on edges, the second has one constraint per edge, and the third is totally unconstrained. For each of them we prove hp error estimates in the H1 norm, optimal with respect to h, the mesh size, and nearly optimal with respect to p, the degree of polynomial approximation. We establish these results for general elements in two and three dimensions. For the unconstrained method, we establish a new approximation result valid on simplicial elements. L2 estimates are also derived for the three methods.

Locally Conservative Coupling of Stokes and Darcy Flows

A locally conservative numerical method for solving the coupled Stokes and Darcy flows problem is formulated and analyzed. The approach employs the mixed finite element method for the Darcy region and the discontinuous Galerkin method for the Stokes region. A discrete inf-sup condition and optimal error estimates are derived.

Multiphysics simulations: Challenges and opportunities

We consider multiphysics applications from algorithmic and architectural perspectives, where “algorithmic” includes both mathematical analysis and computational complexity, and “architectural” includes both software and hardware environments. Many diverse multiphysics applications can be reduced, en route to their computational simulation, to a common algebraic coupling paradigm. Mathematical analysis of multiphysics coupling in this form is not always practical for realistic applications, but model problems representative of applications discussed herein can provide insight. A variety of software frameworks for multiphysics applications have been constructed and refined within disciplinary communities and executed on leading-edge computer systems. We examine several of these, expose some commonalities among them, and attempt to extrapolate best practices to future systems. From our study, we summarize challenges and forecast opportunities.

A discontinuous Galerkin method with nonoverlapping domain decomposition for the Stokes and Navier-Stokes problems

A family of discontinuous Galerkin finite element methods is formulated and analyzed for Stokes and Navier-Stokes problems. An inf-sup condition is established as well as optimal energy estimates for the velocity and L 2 estimates for the pressure. In addition, it is shown that the method can treat a finite number of nonoverlapping domains with nonmatching grids at interfaces.

Agent-based model of inflammation and wound healing : insights into diabetic foot ulcer pathology and the role of transforming growth factor-β1

Inflammation and wound healing are inextricably linked and complex processes, and are deranged in the setting of chronic, nonhealing diabetic foot ulcers (DFU). An ideal therapy for DFU should both suppress excessive inflammation while enhancing healing. We reasoned that biological simulation would clarify mechanisms and help refine therapeutic approaches to DFU. We developed an agent‐based model (ABM) capable of reproducing qualitatively much of the literature data on skin wound healing, including changes in relevant cell populations (macrophages, neutrophils, fibroblasts) and their key effector cytokines (tumor necrosis factor‐α [TNF], interleukin [IL]‐1β, IL‐10, and transforming growth factor [TGF]‐β1). In this simulation, a normal healing response results in tissue damage that first increases (due to wound‐induced inflammation) and then decreases as the collagen levels increase. Studies by others suggest that diabetes and DFU are characterized by elevated TNF and reduced TGF‐β1, although which of these changes is a cause and which one is an effect is unclear. Accordingly, we simulated the genesis of DFU in two ways, either by (1) increasing the rate of TNF production fourfold or (2) by decreasing the rate of TGF‐β1 production 67% based on prior literature. Both manipulations resulted in increased inflammation (elevated neutrophils, TNF, and tissue damage) and delayed healing (reduced TGF‐β1 and collagen). Our ABM reproduced the therapeutic effect of platelet‐derived growth factor/platelet releasate treatment as well as DFU debridement. We next simulated the expected effect of administering (1) a neutralizing anti‐TNF antibody, (2) an agent that would increase the activation of endogenous latent TGF‐β1, or (3) latent TGF‐β1 (which has a longer half‐life than active TGF‐β1), and found that these therapies would have similar effects regardless of the initial assumption of the derangement that underlies DFU (elevated TNF vs. reduced TGF‐β1). In silico methods may elucidate mechanisms of and suggest therapies for aberrant skin healing.

DG Approximation of Coupled Navier-Stokes and Darcy Equations by Beaver-Joseph-Saffman Interface Condition

In this work, we couple the incompressible steady Navier-Stokes equations with the Darcy equations, by means of the Beaver-Joseph-Saffmans condition on the interface. Under suitable smallness conditions on the data, we prove existence of a weak solution as well as some a priori estimates. We establish local uniqueness when the data satisfy additional smallness restrictions. Then we propose a discontinuous Galerkin scheme for discretizing the equations and do its numerical analysis.

A Discontinuous Galerkin Method Applied to Nonlinear Parabolic Equations

Semi-discrete and a family of discrete time locally conservative Discontinuous Galerkin procedures are formulated for approximations to nonlinear parabolic equations. For the continuous time approximations a priori L ∞ (L 2) and L 2(H l) estimates are derived and similarly, l ∞ (L 2) and l 2 (H 1) for the discrete time schemes. Spatial rates in H l and time truncation errors in L 2 are optimal.

A Posteriori error estimates for a discontinuous galerkin method applied to elliptic problems. Log number: R74

Abstract A posteriori error estimates for locally mass conservative methods for subsurface flow are presented. These methods are based on discontinuous approximation spaces and referred to as discontinuous Galerkin methods. In the case where penalty terms are added to the bilinear form, one obtains the nonsymmetric interior penalty Galerkin methods. In a previous work, we proved optimal rates of convergence of the methods applied to elliptic problems. Here, h adaptivity is investigated for flow problems in 2D. We derive global explicit estimators of the error in the L 2 norm and we numerically investigate an implicit indicator of the error in the energy norm. Model problems with discontinuous coefficients are considered.