Iterative Subregion Correction Preconditioners with Adaptive Tolerance for Problems with Geometrically Localized Stiffness

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Communications on Applied Mathematics and Computation

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We present a class of preconditioners for the linear systems resulting from a finite element or discontinuous Galerkin discretizations of advection-dominated problems. These preconditioners are designed to treat the case of geometrically localized stiffness, where the convergence rates of iterative methods are degraded in a localized subregion of the mesh. Slower convergence may be caused by a number of factors, including the mesh size, anisotropy, highly variable coefficients, and more challenging physics. The approach taken in this work is to correct well-known preconditioners such as the block Jacobi and the block incomplete LU (ILU) with an adaptive inner subregion iteration. The goal of these preconditioners is to reduce the number of costly global iterations by accelerating the convergence in the stiff region by iterating on the less expensive reduced problem. The tolerance for the inner iteration is adaptively chosen to minimize subregion-local work while guaranteeing global convergence rates. We present analysis showing that the convergence of these preconditioners, even when combined with an adaptively selected tolerance, is independent of discretization parameters (e.g., the mesh size and diffusion coefficient) in the subregion. We demonstrate significant performance improvements over black-box preconditioners when applied to several model convection-diffusion problems. Finally, we present performance results of several variations of iterative subregion correction preconditioners applied to the Reynolds number


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