Adaptive Finite Elements for Parabolic Partial Differential Equations

抛物型偏微分方程的自适应有限元

• 批准号: 78169562
• 负责人: Professor Dr. Kunibert G. Siebert
• 金额: --
• 依托单位: Institut für Angewandte Analysis und Numerische Simulation (IANS)
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2008
• 资助国家: 德国
• 起止时间: 2007-12-31 至 2011-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/78169562?language=en
• 关键词: Adaptive; Finite; Elements; Parabolic; Partial

During the last years, computer science and scientific computing has become an important research branch located between applied mathematics, applied sciences, and engineering. Nowadays, in numerical mathematics not only simple model problems are treated, but sophisticated and well-founded mathematical algorithms are applied to solve complex problems of real life applications. Usually, the design of such algorithms is based upon analytical properties and a precise numerical analysis of the underlying problem. Real life applications are demanding for computational realization and need suitable and robust tools for a flexible and efficient implementation.Inspired by and parallel to the investigation of real life applications, numerical mathematics has built and improved many modern algorithms which are now standard tools in scientific computing. Examples are adaptive methods, higher order discretizations, fast linear and non-linear iterative solvers, multi-level algorithms, etc. These mathematical tools are able to reduce computing times tremendously and for many applications a simulation can only be realized in a reasonable time-frame using such highly efficient algorithms. This holds especially true for time-dependent parabolic problems that are the main focus of this research project. The aim of the research is the design and analysis of efficient numerical methods for this class of problems. We want to perform basic research on adaptive finite element methods. Nowadays, adaptive methods for stationary problems are well understood and well analyzed. This includes convergence and optimality of practically used algorithms. Adaptive methods for time-dependent problems are still based upon heuristics. Convergence and optimality of such algorithms is still an open problem. Team Siebert mainly focuses on convergence aspects of finite elements which includes improvement of existing estimators and design of new adaptive methods. Team Hoppe analyses control and state constrained optimal control problems for parabolic PDEs and team Funken focuses on time-periodic micromagnetic problems. With the envisaged research project we want to combine the expertise of the three working groups located at the Universities of Augsburg and Ulm. There will be a close cooperation of all teams with respect to a posteriori error analysis and suitable adaptive methods. To our best knowledge, this will be world-wide the only bundled research project with sole focus on adaptive methods for parabolic problems. We expect that the planned interaction of analysis, different discretization methods, and application to various problems leads to new robust and efficient numerical methods also for real life problems.

近年来，计算机科学和科学计算已成为介于应用数学、应用科学和工程之间的一个重要的研究分支。如今，在数值数学中，不仅处理简单的模型问题，而且应用复杂的和有根据的数学算法来解决真实的生活应用的复杂问题。通常，这种算法的设计是基于分析性质和精确的数值分析的基础上的问题。真实的生活应用对计算实现的要求很高，需要合适的、健壮的工具来灵活、高效地实现。受真实的生活应用研究的启发和平行，数值数学已经建立和改进了许多现代算法，这些算法现在是科学计算中的标准工具。例子是自适应方法，高阶离散化，快速线性和非线性迭代求解器，多级算法等，这些数学工具能够大大减少计算时间，对于许多应用程序的模拟只能在合理的时间框架内实现使用这种高效的算法。这对于本研究项目的主要焦点--依赖时间的抛物问题来说尤其如此。研究的目的是设计和分析这类问题的有效数值方法。我们想对自适应有限元方法进行基础研究。如今，平稳问题的自适应方法得到了很好的理解和分析。这包括实际使用的算法的收敛性和最优性。时变问题的自适应方法仍然是基于遗传算法的。这类算法的收敛性和最优性仍然是一个悬而未决的问题。Siebert团队主要专注于有限元的收敛方面，包括改进现有的估计和设计新的自适应方法。Hoppe团队分析抛物型偏微分方程的控制和状态约束最优控制问题，Funken团队专注于时间周期微磁问题。在设想的研究项目中，我们希望将位于奥格斯堡大学和乌尔姆大学的三个工作组的专业知识联合收割机结合起来。所有小组将在后验误差分析和适当的适应性方法方面密切合作。据我们所知，这将是世界范围内唯一的捆绑研究项目，唯一的重点是抛物问题的自适应方法。我们希望，计划的相互作用的分析，不同的离散化方法，并应用到各种问题，导致新的强大和有效的数值方法也为真实的生活问题。