Matrix-free multigrid block-preconditioners for higher order Discontinuous Galerkin discretisations

Efficient and suitably preconditioned iterative solvers for elliptic partial differential equations (PDEs) of the convection-diffusion type are used in all fields of science and engineering. To achieve optimal performance, solvers have to exhibit high arithmetic intensity and need to exploit every form of parallelism available in modern manycore CPUs. The computationally most expensive components of the solver are the repeated applications of the linear operator and the preconditioner. For discretisations based on higher-order Discontinuous Galerkin methods, sum-factorisation results in a dramatic reduction of the computational complexity of the operator application while, at the same time, the matrix-free implementation can run at a significant fraction of the theoretical peak floating point performance. Multigrid methods for high order methods often rely on block-smoothers to reduce high-frequency error components within one grid cell. Traditionally, this requires the assembly and expensive dense matrix solve in each grid cell, which counteracts any improvements achieved in the fast matrix-free operator application. To overcome this issue, we present a new matrix-free implementation of block-smoothers. Inverting the block matrices iteratively avoids storage and factorisation of the matrix and makes it is possible to harness the full power of the CPU. We implemented a hybrid multigrid algorithm with matrix-free block-smoothers in the high order DG space combined with a low order coarse grid correction using algebraic multigrid where only low order components are explicitly assembled. The effectiveness of this approach is demonstrated by solving a set of representative elliptic PDEs of increasing complexity, including a convection dominated problem and the stationary SPE10 benchmark.