Optimal robust solvers for reliable and efficient AFEMs

Many practical physical phenomena are modeled mathematically via partial differential equations (PDEs). Discretizing a PDE via the finite element method (FEM) yields a finite- dimensional problem that can be treated by computers. In particular, safety, environmental, and financial aspects urge numerical simulations to be: reliable (guarantee that a desired accuracy is attained); efficient (require minimal computational cost). Even for symmetric second-order linear elliptic PDEs, the unknown solution may exhibit singularities induced by the given data or the geometry of the domain. Therefore, standard FEMs often fail to achieve a required accuracy. Thus, a refined approach is indispensable: adaptive FEM (AFEM) steers the decrease of the local mesh-size h in the vicinity of potential singularities. In this case, the use of piecewise polynomials of higher order p often leads to yet improved convergence rates but higher computational effort. To attain optimal complexity, the arising discrete problems require optimal iterative solvers. This includes, e.g., a class of multigrid methods ensuring hp- robust contraction: strict error decrease per iteration uniformly with respect to the discretization parameters. Despite many recent developments, unresolved questions include: 1) Can hp-robust solvers be developed for non-symmetric problems? 2) Can an optimal robust solver be designed for non-linear problems? 3) Can the solver itself be made adaptive? How does a non-uniform (element-wise) distribution of p affect the solver contraction? 4) Can such an approach be combined with adaptive mesh-refinement to guarantee (optimal) convergence of the resulting hp-AFEM with iterative solver? Is the convergence parameter-robust, i.e., guaranteed for any user-chosen parameters? The overarching aim of the project is to design and implement optimal, error-contractive solvers for AFEM that are robust with respect to the local mesh-size h and the local polynomial degree p for linear systems stemming from second-order elliptic PDEs. To this end, knowledge and underlying techniques from a-posteriori error analysis, AFEMs, linear algebra, high-order methods as well as implementational and computer science aspects will be necessary. The design and analysis of optimal iterative solvers for hp-adaptive discretizations will substantially push forward our understanding of both adaptivity and iterative methods as well as their interplay. Particular focus is on the hp-robust solvers as well as on parameter- robust convergence and optimality of the overall adaptive algorithms.