Efficient solvers for the implicit time integration of matrix-free high-order methods
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Efficient solvers for the implicit time integration of matrix-free high-order methods


In this thesis, we develop and study efficient solvers for high-order Galerkin methods applied to fluid flow problems.Many flow problems necessitate implicit time-integration schemes in order to be practical. Implicit-in-time discretizations require the solution of nonlinear algebraic systems each time step, which are often in-turn solved by linear solvers. Therefore, the performance of implicit-in-time solvers is largely determined by the performance of the underlying linear solvers.

One approach to create efficient methods is to work with matrix-free operators.Because assembling the underlying discretization matrix can be prohibitively expensive in terms of computational complexity and memory, matrix-free operators are an attractive alternative. These operators replace the matrix-vector products with on-the-fly sum-factorization evaluations of the discretized differential operators instead. Indeed, their high arithmetic intensity makes these operators particularly well suited for modern graphics processing units (GPU) and GPU-accelerated architectures.

These matrix-free operators are particularly challenging to precondition, however, because they by design do not allow access to the underlying matrix entries.We create a suite of efficient matrix-free preconditioners for a range of fluid flow problems that are robust with respect to polynomial degree and mesh size. The main building block solver extends sparse, low-order refined preconditioners with parallel subspace corrections. This work tackles Poisson problems, saddle-point Stokes systems, and the incompressible Navier-Stokes equations in two and three spatial dimensions.

A different set of problems exhibit geometrically localized stiffness, where convergence rates are degraded in a localized subregion of the mesh.Generic preconditioners do not perform well across the entire domain because of mesh size, mesh anisotropy, highly variable coefficients, or more challenging physics in the subregion. Therefore, we seek to save costs by utilizing cheap preconditioners for most of the mesh and only focus our effort on the less expensive subregion problem. Our iterative subregion correction preconditioners correct naive preconditioners with an adaptive inner subregion iteration to reduce the number of costly global iterations. This work demonstrates performance on basic convection-diffusion problems, high Reynolds number compressible flow problems, and a $30^\circ$ angle of attack problem with massively separated flow.

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