Advances in Multilevel Methods for Saddle Point Problems

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In spite of the fast increase in computational power of today's computers, the development of faster algorithms for equational models is paramount for the simulation of phenomena investigated nowadays in practical applications in science and engineering. For a large spectrum of situations in the scientific world, a computer simulation has to replace experimental simulation and data collection. For a computer simulation, in order to obtain accurate approximation of the physical quantities of interest, a powerful computational system has to be combined with a fast and reliable algorithm to test and to confirm the designed models. The project will contribute to the construction and implementation of new and efficient algorithms for solving numerically challenging problems. In particular, the proposed research will focus on constructing fast and robust algorithms for solving computational fluid dynamics and electromagnetic problems. The breakthrough approach of the project is based on recent results in numerical analysis, on new algorithm designing and implementation, and on scientific testing and validation of the new computational tools. The methodology from this project for the time-harmonic Maxwell's equations has a broad range of applications in nano-optics and analog signal packages. The work for solving variational problems has scientific and technical applications in optimization of electrical networks and image restoration. The research will study and develop reliable and efficient numerical algorithms for solving partial differential equations that can be described as variational saddle point systems or mixed variational formulations. The goal of the project is to build robust algorithms for solving such equations in the presence of low regularity of solutions due to discontinuities in data or coefficients. The research will produce a rigorous and systematic analysis of a large class of saddle point problems, focusing on efficient algorithm development and testing. The methods will be based on finite element discretization algorithms in the context of a multilevel and adaptive choice of approximation spaces. The PI's approach uses a new saddle point least-squares type of discretization for systems of PDEs, that takes full advantage of the regularity of the solution and involves an efficient level change criterion that minimizes the running time of the global iterative process. This study will lead to more reliable methods for a variety of applications of the finite element method to science and engineering communities, such as those interested in elasticity, electromagnetism, friction, and computational fluid dynamics. The research findings will be shared within the field and with a more general audience including mathematics and science high-school teachers.