Overlapping Schwarz preconditioners for spectral element discretizations of convection‐diffusion problems

Pavarino, Luca F.

doi:10.1002/nme.327

Cited by 6 publications

(4 citation statements)

References 24 publications

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“…UsingF as a preconditioner allowed FGMRES to obtain convergence rates independent of the mesh size, and mildly dependent of changes in convection strength. Our results compare favorably with those reported in [1], [7], [13] and [19].…”

Section: Variable Wind Resultssupporting

confidence: 89%

“…We demonstrated that this solution method has a weak dependence on Peclet number, and mild dependence on mesh refinement for both h-refinement and p-refinement. Similar conclusions have been reported in [1], [13] and [19] for overlapping and non-overlapping domain decomposition strategies. The use of FDM in our method significantly reduces the computational cost of the subsidiary interior solves throughout the calculation.…”

Section: Variable Wind Resultssupporting

confidence: 87%

“…For our second example, we use a test case from [13] with a variable flow field w = 1 2 ((1−x 2 )(1+y), x((1+y) 2 −4)). Boundary …”

Section: Variable Wind With Curved Open Streamlinesmentioning

confidence: 99%

“…We emphasize the case where Pe is large, although these techniques apply to diffusion-dominated flows as well. Similar ideas are explored in [13] for stabilized spectral element discretizations via overlapping domain decomposition. Our method is based on non-overlapping domain decomposition and takes advantage of Fast Diagonalization to eliminate degrees of freedom in element interiors.…”

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confidence: 98%

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Fast iterative solver for convection‐diffusion systems with spectral elements

Lott

Elman

2011

Numerical Methods Partial

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Abstract. We introduce a solver and preconditioning technique based on Domain Decomposition and the Fast Diagonalization Method that can be applied to tensor product based discretizations of the steady convection-diffusion equation. The method is based on iterative substructuring where fast diagonalization is used to efficiently eliminate the interior degrees of freedom and subsidiary subdomain solves. We demonstrate the effectiveness of this method in numerical simulations using a spectral element discretization.Key words. Convection-Diffusion, Domain Decomposition, Preconditioning, Spectral Element Method 1. Introduction. Numerical simulation of fluid flow allows for improved prediction and design of natural and engineered systems such as those involving water, oil, or blood. The interplay between inertial and viscous forces in a fluid flow dictates the length scale where energy is transferred, thus determining the resolution required to capture flow information accurately. This resolution requirement poses computational challenges in situations where the convective nature of the flow dominates diffusive effects. In such flows, convection and diffusion occur on disparate scales, causing sharp flow features that require fine numerical grid resolution. This leads to a large system of equations which is often solved using an iterative method. Exacerbating the challenge of solving a large linear system, the discrete convection-diffusion operator is non-symmetric and poorly conditioned. This leads to slow convergence of iterative solvers. In total, as convection dominates the flow the discrete fluid model becomes exceedingly challenging to solve.In recent years, the spectral element method has gained popularity as a technique for numerical simulation of fluids [9], [18]. This is due in part to the method's high-order accuracy, which produces solutions with low dissipation and low dispersion with relatively few degrees of freedom. Also important is the inherent computational efficiency gained through the use of a hierarchical grid structure based on unstructured macro-elements with fine tensor-structured interiors. This structure has enabled the development of efficient multi-level solvers and preconditioners based on Fast Diagonalization and Domain Decomposition [8], [10], [21]. Application of these techniques, however, has been restricted to symmetric systems.One way to apply such methods to non-symmetric systems is through use of timesplitting techniques, which split the system into symmetric and non-symmetric components. For convection-diffusion systems, the standard method for performing steady and unsteady flow simulations with spectral elements is operator integration factor splitting (OIFS) [12], which requires time integration even in steady flow simulations. Using this standard approach, convection and diffusion are treated separately; convection components are tackled explicitly using a sequence of small time steps that satisfy a CFL condition, and diffusive components are treated implicitly with larger t...

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