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A-posteriori-steered h- and p-robust multigrid solvers and optimal complexity of AFEM
Abstract
Adaptive Finite Element Methods (AFEMs) drive the local mesh refinement capturing potential singularities of the (unknown) PDE solution. The aim is to achieve optimal convergence rates with respect to the dimension of the finite element space, which is determined by the polynomial degree p and the local mesh size h. However, the adaptive algorithm is inherently cumulative in nature: solutions are computed on consecutive meshes before a desired accuracy can be ensured. Thus, in practice, one strives instead to achieve optimal complexity, i.e., optimal rate of convergence with respect to the overall computation cost. To this end, in the context of second order elliptic PDEs, we: (i) design multigrid iterative solvers that ensure linear complexity and uniform contraction; (ii) employ an adaptive stopping criterion which allows to discern and balance error sources; (iii) employ nested iteration, i.e., the last computed solver-iterate is the initial guess in the newly-refined mesh. The adaptive algorithm is shown to be unconditionally convergent and of optimal complexity for sufficiently small adaptivity parameters.
