Augmented Mixed Finite Element Methods: Analysis, Applications, and Comparison with Least-Squares Methods
DescriptionIn this project, we plan to study a special Galerkin/Least-Squares method, the augmented mixed finite element method, and its relation to the standard least-squares finite element methods (LSFEM). The LSFEM has many known advantages, it is automatically stable and optimal, the first-order system is physically meaningful, it has a natural and accurate a posteriori error estimator, and it is ideal to be used as a cost function in developing neuron network based algorithms. But the full least-squares system is often strongly coupled without integration by parts and there is little room to play with the formula. Thus, it is often hard to do certain analysis. Besides the full bona fide LSFEM, another idea to apply the LS philosophy is the so-called Galerkin/Least-Squares (GaLS) method. The GaLS method is a method combing the LS and Galerkin methods. Some LS terms are added to the original variational formulation to enhance the stability in GaLS methods. In this project, we consider a very special GaLS method, the augmented mixed finite element method. The central idea of the augmented mixed method is that someconsistent terms (usually from constitutive equations and other relations of different physical quantities) are added as the difference of two squares instead of the usual least-squares terms. Other terms (usually from equilibrium equations) can still be added to the mixed weak problem in the least-squares forms to enhance the stability. As a partial LS method, the augmented mixed finite element method shares many properties with the classic LSFEM, it is also stable and approximates the physical quantities in their native spaces. Furthermore, since we only use LS principle partially, the system is more flexible and can have some properties that the full LSFEM does not have. In this project, we will study the robust and local optimal error analysis for augmented mixed finite element method, which is extremely important for problems depending on parameters. We will also study the goal-oriented a posteriori errorestimates based on adjoints problem, which has many applications in science, engineering, and technology. Stokes problems with slip boundary conditions will also be studied. We expect that the proposed research will result in new insights and develop better numerical methods for computationally challenging problems. The results of the proposed research will provide powerful computational tools for many problems of mathematical interests and practical applications.
|Effective start/end date||1/01/23 → …|