Computational nonlinear PDEs

The ultimate goal of any numerical scheme is to compute a discrete solution with error below a prescribed tolerance at, up to a multiplicative constant, the minimal computational cost. Usually, the convergence behavior of numerical schemes is, however, spoiled by singularities of the given data and/or the unknown solution. Moreover, nonlinear PDEs naturally lead to discrete nonlinear systems, which cannot be solved exactly, but have to be solved approximately by (usually even nested) iterative methods. Therefore, the solver has to balance the discretization error as well as the solver error by controlling and steering (1) the underlying discretization, (2) the error from linearizing the discrete nonlinear systems, (3) the algebraic error from the inexact solution of the arising linear equations. While there is a rich body on a-posteriori error estimation for nonlinear PDEs, which also includes the inexact solution of the (nonlinear and linear) discrete systems, the thorough mathematical understanding of optimal convergence behavior of the related adaptive strategies is still in its infancy. The proposed research aims to provide a sound mathematical foundation of adaptive algorithms for nonlinear model problems. While available results on rate optimality of adaptive algorithms usually focus on algebraic convergence rates with respect to the degrees of freedom, the important punchline of the proposed research will be the mathematical understanding of optimal rates with respect to the overall computational costs. We believe that this question is timely and of utmost importance to practitioners working in the field of computational nonlinear PDEs. All theoretical findings will be implemented in MATLAB where the codes will be provided to the academic public online.

Many physical phenomena are modeled by partial differential equations. In practical applications, however, these equations can rarely be solved exactly. Instead, their solutions must be approximated numerically using computers. This typically involves several steps: first, the differential equation is discretized; second, the resulting nonlinear discrete problem is linearized; and third, the arising linear systems of equations are solved efficiently. Each of these steps introduces its own source of error, in particular discretization errors, linearization errors, and solver errors. For reliable and efficient simulations, it is essential to balance these different error contributions appropriately. Within the framework of the project, computable a-posteriori error estimators and adaptive algorithms were developed to automatically control this balance. It was mathematically proven that, for certain classes of practically relevant model problems, these algorithms converge with optimal complexity. In other words, the error of the computed numerical approximation decreases in a quasi-optimal way with respect to the computational time required for the simulation.