Adaptive Uzawa-type FEM for nonlinear PDEs

Many processes in nature, science, and engineering are mathematically modeled by so- called partial differential equations (PDEs). Even though it is often possible to prove the existence and uniqueness of a solution for a given PDE, in most cases this solution cannot be expressed in a closed analytical form. Therefore, numerical methods are required to approximate the desired solution and to understand its behavior quantitatively. The goal of any numerical method is to compute a discrete solution that approximates the unknown exact solution up to a prescribed tolerance with minimal computational effort, meaning that the approximation error should decrease optimally with respect to computing time. Within this project, adaptive algorithms will be developed and mathematically analyzed that meet these requirements. This involves controlling and balancing the various sources of error in numerical simulations in a mathematically rigorous way. The specific focus of the project is on nonlinear and coupled differential equations and follows a strategy inspired by the Uzawa algorithm for the Stokes equations: Linearization at the continuous level of the differential equations, Discretization of the linearized equations using the finite element method (FEM), Numerical solution with suitable iterative solvers for linear systems. This approach is particularly advantageous when dealing with coupled problems, as it allows the equations to be decoupledthat is, instead of solving one large (possibly nonlinear) coupled system of equations, two smaller, independent linear systems need to be solved.