Mathematical Aspects of Discontinuous Galerkin Methods (Mathématiques et Applications Book 69)

This book introduces the basic ideas to build discontinuous Galerkin methods and, at the same time, incorporates several recent mathematical developments. The presentation is to a large extent self-contained and is intended for graduate students and researchers in numerical analysis. The material covers a wide range of model problems, both steady and unsteady, elaborating from advection-reaction and diffusion problems up to the Navier-Stokes equations and Friedrichs' systems. Both finite element and finite volume viewpoints are exploited to convey the main ideas underlying the design of the approximation. The analysis is presented in a rigorous mathematical setting where discrete counterparts of the key properties of the continuous problem are identified. The framework encompasses fairly general meshes regarding element shapes and hanging nodes. Salient implementation issues are also addressed. This book introduces the basic ideas for building discontinuous Galerkin methods and, at the same time, incorporates several recent mathematical developments. It is to a large extent self-contained and is intended for graduate students and researchers in numerical analysis. The material covers a wide range of model problems, both steady and unsteady, elaborating from advection-reaction and diffusion problems up to the Navier-Stokes equations and Friedrichs' systems. Both finite-element and finite-volume viewpoints are utilized to convey the main ideas underlying the design of the approximation. The analysis is presented in a rigorous mathematical setting where discrete counterparts of the key properties of the continuous problem are identified. The framework encompasses fairly general meshes regarding element shapes and hanging nodes. Salient implementation issues are also addressed This Text Introduces The Basic Ideas To Build Discontinuous Galerkin Methods And, At The Same Time, Incorporates Several Recent Mathematical Developments. The Presentation Is To A Large Extent Self-contained And Is Intended For Graduate Students And Researchers In Numerical Analysis. Pt. 1. Scalar First-order Pdes -- Pt. 2. Scalar Second-order Pdes -- Pt. 3. Systems. Daniele Antonio Di Pietro, Alexandre Ern. Includes Bibliographical References (p. 355-374) And Indexes. Front Matter....Pages i-xvii Basic Concepts....Pages 1-34 Front Matter....Pages 35-35 Steady Advection-Reaction....Pages 37-65 Unsteady First-Order PDEs....Pages 67-115 Front Matter....Pages 117-117 PDEs with Diffusion....Pages 119-186 Additional Topics on Pure Diffusion....Pages 187-237 Front Matter....Pages 239-239 Incompressible Flows....Pages 241-291 Friedrichs’ Systems....Pages 293-341 Back Matter....Pages 343-384