Discontinuous Galerkin methods for solving elliptic and parabolic equations - theory and implementation
Discontinuous Galerkin methods for solving elliptic and parabolic equations - theory and implementation
复制标题
DOI：
10.1137/1.9780898717440
复制
发表时间：
2008-12
期刊：
影响因子：
--
通讯作者：
B. Rivière
中科院分区：
文献类型：
--
作者：
B. Rivière
求解偏微分方程的间断Galerkin（DG）方法是在20世纪90年代后期发展起来的，已经成为计算科学家们的热门。这本书涵盖了理论和计算，因为它侧重于三个原始DG方法-对称内部惩罚Galerkin，不完整内部惩罚Galerkin和非对称内部惩罚Galerkin这是内部惩罚方法的变化。作者提供了基本的分析工具，并讨论了编码问题，包括数据结构，局部矩阵的构造，和组装的全球矩阵。还包括计算实例和重要工程问题的应用。间断Galerkin方法求解椭圆和抛物方程：理论和实现分为三个部分：第一部分着重于DG方法在一维和高维二阶椭圆问题中的应用。第二部分介绍了含时抛物型方程的无对流和有对流两种情况。第三部分包含DG方法在固体力学（线弹性）、流体动力学（斯托克斯和纳维斯托克斯）和多孔介质流动（两相和混溶驱替）中的应用。附录包含证明和MATLAB代码的一维问题的椭圆方程和例程写在C中，对应于算法的实施DG方法在二维或三维。观众：这本书是为数值分析师，计算和应用数学家感兴趣的数值方法偏微分方程或谁研究的应用程序中讨论的书，工程师谁在流体动力学和固体力学工作，并希望使用DG方法为他们的数值结果。这本书适合于有限元法、偏微分方程数值方法、数值分析和科学计算的研究生课程。第一章是适合高年级本科班的科学计算。内容包括：图表清单;表格清单;算法列表;序言;第一部分：椭圆问题;第一章：一维问题;第二章：高维问题;第二部分：抛物问题第三章：抛物线问题第四章：对流的抛物问题;第三部分：应用;第5章：线性弹性;第6章：斯托克斯流;第7章：纳维尔-斯托克斯流;第8章：多孔介质中的流动;附录A：求积规则;附录B：DG代码;附录C：近似结果;参考书目;索引。
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.