Rate Optimality of Adaptive Finite Elements for Parabolic Partial Differential Equations

在这个项目中，我们主要研究抛物型偏微分方程的有限元方法，主要集中在自适应离散上。自然科学和工程中的许多过程都是用抛物型偏微分方程组来模拟的。这类问题的数值分析是数值数学前沿的一个非常活跃的研究领域。对于许多现实生活中的问题，基于统一方法的计算过于耗时。为了有效地对暂态问题进行数值模拟，需要具有最佳复杂性的自适应方法。我们的经验表明，椭圆偏微分方程组的自适应有限元方法已经相当成熟。它们在许多实际应用中的应用显示出非常健壮的性能。此外，还提供了关于自由度的收敛和最优衰减率的证明。这些证明是基于对自适应方法作为一个整体的非常精确的数学理解。今天的愿景是能够获得类似的稳健和有效的自适应方法来处理pPDE。对于实际应用的自适应仿真环境的设计将极大地得益于对pPDE自适应有限元的更好的理解和更好的数学基础。然而，从我们的角度来看，对这些方法的整体分析还处于初级阶段。我们只有一个热方程的收敛结果；最优性能的证明是完全缺失的。因此，建议的研究项目集中在对偏微分方程组的速率最优自适应有限元方法的设计和分析上。这一重要而具有挑战性的课题需要对适当的有限元离散的基本性质进行大量的研究。我们将研究自适应生成的离散解序列的自适应逼近类和衰减率。后者可能需要新的网格和时间步长自适应方案。通过所提出的研究，我们将在很大程度上促进适合于高性能计算机上实际应用的pPDE的最优自适应方法的开发和数值分析。此外，这些研究将有助于更好地理解强非对称问题自适应方法的最优性分析所需的知识。

In this project we want to analyze finite element methods for parabolic partial differential equations (pPDEs) with main focus on adaptive discretizations.Many processes in natural science and engineering are modeled by pPDEs. Numerical analysis for this problem class is a very active research area at the forefront of numerical mathematics. For many real life problems computations based on uniform methods are by far too time con- suming. The efficient numerical simulation of transient problems necessitates adaptive methods with optimal complexity.Our experience shows that adaptive finite elements for elliptic partial differential equations (ePDEs) have gained substantial maturity. Their use in numerous practical applications reveals a very robust performance. Moreover, proofs of convergence and optimal decay rates in terms of degrees of freedom (DOFs) are available. These proofs are based on a very precise mathematical understanding of the adaptive method as a whole.It is a todays vision to have access to similar robust and efficient adaptive methods for pPDEs. The design of adaptive simulation environments for real life applications would tremendously profit from a better understanding and a better mathematical foundation of adaptive finite elements for pPDEs. However, the analysis of such methods as a whole is yet, from our point of view, in its infancy. We have a single convergence result for the heat equation; a proof of an optimal performance is missing completely.The proposed research project therefore focuses on the design and analysis of rate optimal adaptive finite element methods for pPDEs. This important and challenging topic requires a substantial amount of investigations concerning fundamental properties of appropriate finite element discretizations. We will investigate adaptive approximation classes and decay rates of adaptively generated sequences of discrete solutions. The latter may require new mesh and time-step adaptation schemes.With the proposed research we will substantially foster the development and numerical analy- sis of optimal adaptive methods for pPDEs suitable for real-life applications on high performance computers. In addition, the investigations will contribute to a better understanding needed for the optimality analysis of adaptive methods for strongly non-symmetric problems.