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Efficient solvers for the implicit time integration of matrixfree highorder methods
 Franco, Michael
 Advisor(s): Persson, PerOlof
Abstract
In this thesis, we develop and study efficient solvers for highorder Galerkin methods applied to fluid flow problems.Many flow problems necessitate implicit timeintegration schemes in order to be practical. Implicitintime discretizations require the solution of nonlinear algebraic systems each time step, which are often inturn solved by linear solvers. Therefore, the performance of implicitintime solvers is largely determined by the performance of the underlying linear solvers.
One approach to create efficient methods is to work with matrixfree operators.Because assembling the underlying discretization matrix can be prohibitively expensive in terms of computational complexity and memory, matrixfree operators are an attractive alternative. These operators replace the matrixvector products with onthefly sumfactorization 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 GPUaccelerated architectures.
These matrixfree 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 matrixfree 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, loworder refined preconditioners with parallel subspace corrections. This work tackles Poisson problems, saddlepoint Stokes systems, and the incompressible NavierStokes 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 convectiondiffusion problems, high Reynolds number compressible flow problems, and a $30^\circ$ angle of attack problem with massively separated flow.
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