Robust adaptivity for nonlinear partial differential equations

Partial differential equations (PDEs) are ubiquitous in the modelling of real-world problems. The exact solution of such an equation can almost never be given in closed form, so that numerical schemes are required to at least approximate it. The ultimate goal of any such numerical scheme is to compute a discrete approximation with error below some desired tolerance at the expense of minimal computational cost. 

 To this end, it is necessary to accurately quantify the overall error and identify its different components. The first essential component is the discretization error, which stems from approximating the sought-after PDE solution by functions in a finite-dimensional space, typically piecewise polynomials of some fixed degree on some mesh of the considered computational domain. To decrease this error component, one can enrich the employed space, for instance by mesh refinement. While it may suffice to refine the current mesh uniformly in case of a smooth solution, singularities have to be locally resolved in case of a non-smooth solution to guarantee minimal computational cost of the scheme. Discretization of nonlinear PDEs naturally lead to nonlinear discrete systems, which cannot be solved exactly, and thus have to be (iteratively) linearized. This yields the second essential error component, the linearization error, which can be decreased by applying an additional step of the employed iterative linearization solver. Finally, even the solutions of the linearized discrete systems can only be computed approximately, as their exact (up to rounding errors) computation by a direct solver would be prohibitively expensive. The corresponding third essential error component, the algebraic error, can be typically decreased by applying an additional step of the employed iterative algebraic solver. 

 To achieve the aforementioned goal, it is crucial to balance all involved error components. Since none of these error components is computable exactly and/or in an inexpensive way, a practical numerical algorithm has to accurately estimate them and then balance the corresponding a posteriori computable error estimators.

 The proposed research project aims to break new ground in the mathematical understanding of such adaptive algorithms.