Rate Optimality of Adaptive Finite Elements for Parabolic Partial Differential Equations

• Applicant: Professor Dr. Kunibert G. Siebert
• Subject Area: Mathematics
• Term: from 2015 to 2019
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 273218570

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.

• DFG Programme: Research Grants