Generalized preconditioned conjugate gradients for adaptive FEM with optimal complexity

Abstract: We consider adaptive finite element methods (AFEMs) with inexact algebraic solver for second-order symmetric linear elliptic diffusion problems. We formulate and analyze a non-linear and non-symmetric geometric multigrid preconditioner for the generalized preconditioned conjugate gradient method (GPCG) used to solve the arising finite element systems. Moreover, a linear and symmetric variant of the geometric multigrid preconditioner that is suitable for the (standard) preconditioned conjugate gradient method (PCG) is provided and analyzed. We show that both preconditioners are optimal in the sense that, first, the resulting algebraic solvers admit a contraction factor that is independent of the local mesh size h and the polynomial degree p, and, second, that they can be applied with linear computational complexity. Related to this, quasi-optimal computational cost of the overall adaptive finite element method is addressed. Numerical experiments underline the theoretical findings.