Adaptive Discontinuous Galerkin Methods with Applications

自适应间断伽辽金方法及其应用

• 批准号: 288315-2013
• 负责人: Schötzau, Dominik
• 金额: $ 1.38万
• 依托单位: University of British Columbia
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=571391
• 关键词: Adaptive; Discontinuous; Galerkin; Methods; Applications

Many problems of engineering interest are posed in complex, possibly non-smooth and three-dimensional domains. They often involve multiple physical models and scales, as well as highly varying or non-linear material properties. Advanced numerical techniques are required for meaningful predictions. To achieve improved and high-order accurate simulation outputs, we shall focus on the development and implementation of discontinuous Galerkin (DG) finite element methods, combined with adaptive refinement strategies. In recent years, these methods established themselves as robust and flexible simulation techniques for wide classes of problems in computational fluid dynamics, structural mechanics and electro-magnetics. They are naturally suited for adaptive computations, and for providing a unified computational platform for simulating coupled multi-physics models in three space dimensions. The objectives of this proposal are twofold. First, we plan to design and analyse adaptive discontinuous Galerkin (DG) finite element methods for the numerical approximation of partial differential equations of various types in three space dimensions. We are particularly interested in realizing fully automatic refinement strategies, which allow for anisotropic refinements in both the elemental mesh sizes and approximation degrees, and which yield high-order or exponential convergence rates in the numbers of degrees of freedom, also when singularities or layers are present in the phenomena of interest. In addition, we wish to develop fast and efficient spectral/hp element domain decomposition solvers. The second objective is the further advancement and efficient implementation of these methods as applied to coupled incompressible fluid flow problems. We intend to continue our work on fast solvers for electrically conducting fluids interacting with magnetic fields, and to investigate similar techniques for the computer simulation of non-isothermal incompressible flow problems.

许多工程上感兴趣的问题都是在复杂的、可能是非光滑的三维域中提出的。它们通常涉及多个物理模型和比例，以及高度变化或非线性的材料特性。为了进行有意义的预测，需要先进的数值技术。为了实现改进和高阶精度的模拟输出，我们将专注于不连续伽辽金（DG）有限元方法的开发和实施，结合自适应细化策略。近年来，这些方法确立了自己作为强大的和灵活的模拟技术，在计算流体动力学，结构力学和电磁学的广泛类的问题。它们自然适合于自适应计算，并为模拟三维空间中的耦合多物理场模型提供统一的计算平台。 这项建议有两个目的。首先，我们计划设计和分析自适应间断伽辽金（DG）有限元方法的数值逼近的各种类型的偏微分方程在三维空间。我们特别感兴趣的是实现全自动的细化策略，它允许各向异性细化的基本网格尺寸和近似度，并产生高阶或指数收敛率的自由度的数量，也当奇点或层存在于感兴趣的现象。此外，我们希望开发快速有效的谱/HP元素区域分解求解器。第二个目标是进一步的进步和有效的实施，这些方法适用于耦合不可压缩流体流动问题。我们打算继续我们的工作，快速求解器的导电流体与磁场相互作用，并探讨类似的技术，非等温不可压缩流问题的计算机模拟。