Browsing by Author "Thiele, Christopher"

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    Inexact Hierarchical Scale Separation for Linear Systems in Modal Discontinuous Galerkin Discretizations
    (2018-04-20) Thiele, Christopher; Riviere, Beatrice
    This thesis proposes the inexact hierarchical scale separation (IHSS) method for the solution of linear systems in modal discontinuous Galerkin (DG) discretizations. Like p-multigrid methods, IHSS alternates between discretizations of different polynomial order to improve the computational performance of solving linear systems. IHSS uses two discretizations, which are obtained from a hierarchical splitting of the modal DG basis, resulting in two weakly coupled problems for the low-order and high-order components of the solution (coarse and fine scale). While a global linear system of reduced size is solved for the coarse-scale problem, the fine-scale components are updated locally. IHSS extends the original hierarchical scale separation method, using an iterative solver to approximate the coarse-scale problem and shifting more work to the highly parallel local fine-scale updates. Convergence and computational performance of IHSS are evaluated using example problems from an application in the oil and gas industry, the simulation of the phase separation of binary fluid mixtures in three spatial dimensions. The problem is modeled by the Cahn–Hilliard equation, a fourth-order, nonlinear partial differential equation, which is discretized using the nonsymmetric interior penalty DG method. Numerical experiments demonstrate the applicability of IHSS to the linear systems arising in this problem. It is shown that their solution can be significantly accelerated when common iterative methods are used as coarse-scale solvers within IHSS instead of being applied directly. All parameters of the method are discussed in detail, and their impact on computational performance is evaluated.
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    Iterative Methods and Multiscale Methods for Linear Systems in Modal Discontinuous Galerkin Discretizations
    (2021-04-28) Thiele, Christopher; Riviere, Beatrice
    Iterative methods for the solution of linear systems are a core component of many scientific software packages, especially of numerical simulations in which the discretization of partial differential equations in two or three dimensions and with high spatial resolution often results in large, sparse linear systems. Since their introduction decades ago, iterative approaches such as the conjugate gradient method, GMRES, and BiCGStab have remained popular and relevant, and they have proven themselves to be scalable tools in eras of exponential growth in computing power and increasing heterogeneity of computing hardware. In this thesis, I evaluate the convergence and computational performance of iterative solvers for linear systems obtained from modal discontinuous Galerkin (DG) discretizations. Specifically, I focus on systems that arise in the simulation of pore-scale multi-phase flow. Besides standard Krylov subspace methods with algebraic preconditioners, the evaluation focuses on multigrid methods, which, in their more algebraic variants, can also be viewed as iterative linear solvers. In particular, I discuss how p-multigrid methods, which use discretizations of different order instead of discretizations on different meshes, assume a simple algebraic structure for modal DG discretizations. I then show how hierarchical scale separation (HSS), a recently proposed multiscale method for modal DG discretizations, can be incorporated into the p-multigrid framework, and I discuss the unified implementation of both methods in highly parallel computing environments. I analyze how the computational performance of these methods is affected by their tunable parameters, and I demonstrate in numerical experiments that properly calibrated p-multigrid methods can accelerate large pore-scale flow simulations significantly. Moreover, I show how the main ideas of HSS techniques can be used to further accelerate p-multigrid methods.