(untitled) (Turkish Journal of Computer and Mathematics Education (TURCOMAT), 2018)
The numerical approximation of partial differential equations (PDEs) plays a vital role in many scientific and engineering applications. Mixed finite element methods have emerged as powerful techniques for solving a wide range of PDEs due to their ability to handle problems with mixed variables, such as fluid flow and elasticity. However, ensuring the convergence of mixed finite element approximations remains a challenging task. It presents a comprehensive analysis of the optimization strategies employed to enhance the convergence of mixed finite element approximations. We investigate the key factors that impact the convergence behaviour of these methods and propose various techniques to mitigate convergence issues. we discuss the importance of appropriate discretization strategies for mixed finite element approximations. We analyze the impact of element types, mesh refinement, and stabilization techniques on the convergence rates. By examining the properties of the underlying mixed variational formulations, we identify the optimal discretization choices that lead to improved convergence behaviour. We delve into the analysis of numerical stability and consistency in the context of mixed finite element methods. We explore the role of stabilization techniques, such as bubble functions and penalty terms, in mitigating instabilities and achieving optimal convergence rates. We investigate the effect of different stabilization parameters and establish guidelines for their selection to ensure both stability and convergence. We address the issue of error estimation and adaptivity in mixed finite element approximations. We review error indicators and adaptive mesh refinement strategies that enable the refinement of regions with high solution gradients, thus enhancing the convergence rates. We discuss the interplay between error estimation and adaptive refinement and present numerical examples illustrating their effectiveness. We highlight recent advancements in optimization algorithms specifically tailored for enhancing convergence in mixed finite element approximations. We explore strategies like multigrid methods, preconditioning techniques, and domain decomposition methods, which accelerate the convergence rates and enable the solution of large-scale problems. This paper provides a comprehensive analysis of the optimization of convergence for mixed finite element approximations. It serves as a valuable resource for researchers and practitioners seeking to improve the efficiency and accuracy of numerical solutions obtained through mixed finite element methods.